If you want regulated 48v DC, the actual voltage must be a bit higher than that, at least by 1-2 volts. So, after the bridge rectifier and that capacitor (the 470uF in your picture), you need to have at least 50v.
The transformer outputs AC, which when rectified using those diodes creates a DC output with a peak voltage equal to about 1.414 x Vac .
Also, there will be a bit of voltage loss on the diodes inside the bridge rectifier, usually about 1-1.5v.
Also, remember it's only a peak DC voltage of 50v, what comes from the transformer are still some waves of voltage going up and down, and that's why that capacitor is there in the circuit - when the wave goes up towards 50v it fills up and when the wave goes down towards 0v, the capacitor gives the needed energy to maintain the voltage constant. The capacitor should be picked to keep the voltage above the minimum required to keep the 48v regulator happy, which is 50v.
So, let's say 50-52v is what you want, in which case you add those 1-1.5v lost in the bridge rectifier and you have a maximum of about 54v. Now if you divide this by 1.414 you know you need a transformer that outputs around 38v AC.
In most cases, you'll be fine going with a standard 36v AC transformer or one with two 18v secondary windings that you can put in series.
That capacitor should definitely be at least rated for 63v, if you're dabbling with 48v+.
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To regulate the 48v, there are plenty of linear regulators which can tolerate a high voltage at the input, for example LM317 with
HV (or similar endings in the name) can work with up to 60v. So you just use two resistors to adjust the output of this linear regulator to 48v and you're done.
Here's some suggestions, just pick the one that can do your output current :
http://uk.farnell.com/webapp/wcs/stores/servlet/Search?catalogId=15001&langId=44&storeId=10151&categoryId=700000004335&sort=P_PRICE&pageSize=25&showResults=true&aa=true&pf=110421493,110421580,111680855,111680864,111680866&min=110421580 To get 5v, it's kind of a bad idea to use a linear regulator because lin. regs. dissipate the difference between input and output voltage as heat.. so for example at 100mA, you'd produce (48v-5v)x0.1A= 4.3w of heat.
You can buy dc-dc converters from eBay that can work with high voltage (the standard ones usually can only handle up to about 40v), you just have to use the right keywords. Or, you could pick up a switching regulator chip and look in the datasheet and buy the other components needed and make yourself a switching regulator for 5v.
Alternatively, if you buy a transformer with two secondary windings, then you could use one and put its own bridge rectifier and capacitor and you'll have the more manageable 25-30v and then you can use really cheap dc-dc switching regulaor chips like MC34063 (dave did videos on how to use one and there's online calculators for it) or you can buy one of those car cigarette usb chargers and pull out the insides and use them right away.
No, it's not a smart idea to use resistors as voltage dividers, it can make the resistors very hot (and they're small and not easy to cool) and it generally doesn't work right.