PSU design (major revisions): Transformer calculations help?

Recently I have attempting to design a proper dual-rail power supply that will allow me to set a voltage as low as +-1V up to +-30V in 0.1V increments at (hopefully) 3 significant digits (at least for the lower voltage settings). Anyway, this supply is also going to be current limited to up to 5A,again, it can be set to just about anything. I plan on using an Arduino micro-controller to set the output. In order to do this, I plan on using the analogWrite functions, or better yet, a legit DAC. There will be 4 outputs from the Arduino that will set the power supply output by applying a 0-5V voltage on the input of the 2 current limits and 2 voltage sets. (one for the negative rail, one for the positive).

However, I have kept running into the same problem: how do I plan on driving this linear power supply with up to 200W*?

My first idea was to use a a MOT, due to their high-power capabilities, and re wind the secondary with the right number of turns to achieve this output. However, I have heard that these transformers are not optimal for continuous running due to their poor and cheap design. (losses are very high). My second idea was to search around for a 250VA transformer. However, even until now, the VA rating confuses me.

How does VA compare to W? I know this has something to due with reactive power, real power, and apparent power. However, I have no intuition of any of these 'powers.' How would I go about calculating the correct size transformer for the job, also, I am going to assume this linear power supply has the properties of a resistive load, since it is rectified and smoothed with a filter capacitor, so practically nothing should react with the AC power. (unless there is something more to the full-bridge rectifier setup I am considering.)

This is when I came across unwound toroidal cores found on eBay for $25, the perfect price range! However, this has raised more questions! to start off, beyond turns ratio, I do not know now many turns I need for the AC side of things. I know intuitively and from experience, mains-frequency transformers do not work with only one (or even few) winding(s). I think this has to do with saturation, but I'm no expert by any means. and the inductive reactance of the transformer's primary. How do I calculate losses, inductance, and other important parameters of a homemade transformer like this?

Things get very nasty when I look back at rewinding an old transformer. Now I have all these questions about inductive reactance, power, currents, magnetic flux and saturation, but also, about determining the original power rating of something like a very old small welding transformer or one from a large 10A car-battery charger. Is it possible to approximate the power by measuring the dimensions of the core? How close will this approximation be? 

After getting frustrated with this, I considered alternative approaches. What if I purchased 2 ~20V ~6A SMPS (switch mode power supplies) connected them in series, and connect the center tap of my linear supply to the joining point between the 2 SWPS's? Would this be unstable and be bad for the SMPS if a load was connected between the 'outputs' of this new center tapped supply? Would any sort of balancing be required?

Also, a bigger problem includes how this will be connected to my linear PSU design. With a low voltage @ high currents, I would be wasting a LOT of power, power that has to be dissipated away from the transistors. This heat can approach 200W, which is company unreasonable! Anyway, I would them have to either a switching preregulator, or modify the SMPS's so the voltage can be controlled easily and varied between, say, 3V to 20V. absolute accuracy is not required, close enough, and rest of my PSU should handle it.

This becomes seemingly impractical too, and many other considerations need to be made. What should I do? what are the calculations and factors I need to know? i do not have an LCR meter to measure inductance, so trial and error is out. Does anyone here have experience at this? Help would be greatly appreciated!

*The 200W figure was calculated by taking 40V, (What I believe would be a safe to allow some slack for +-5V voltage drop across my 2 shunts and transistors) and multiplying it to 5A of current for the maximum power output.
I have added an image of my current design, and I have modularized it the best I could. The YELLOW is all my current power-management circuitry. Currently just a transformer with many taps, going to a currently-undesigned switch box that will change the voltage on the output, which is then rectified and enters a filtering capacitor, finally entering the circuit. 

The GREEN field is the voltage set. It is the most major part of the PID feedback loop, along with the ORANGE field. It works simply by feeding a voltage to the positive of a op amp configured as a comparator, and with negative feedback from the output. It then outputs a signal to the transistor, turning it either more ON, or more OFF depending on how the output voltage compares to the +Vset. The negative portion is largely the same, but the input voltage needs to be inverted so the output voltage is set negative properly. I was not able to use less than 2 op amps for this portion, unfortunately.

The ORANGE field is current set. It works by measuring the voltage drop across the shunt resistor, and outputting a unity voltage that is referenced to ground, instead of to the positive rail. (It took me forever to finalize and perfect that!!!) Anyway, this voltage is then fed into a op-amp configured as a comparator to drive the transistor. The BLUE field is my switching regulation topology, which is controlled by both the ORANGE and GREEN fields. Do you like my use of diodes as a super-simple voltage or current selection switch? the op amp that outputs a lower voltage is the one that gets 'listened to' by the transistors. This way, current and voltage mode enable properly. This does add a small problem when it comes to powering the op amps, all of them have to be powered off of slightly higher voltages to swing the full range due to the voltage drops of those diodes.

In the PINK field is simply a single-transistor solution to a constant current load. This allows the regulator to be regulated even at very low voltage set levels. This is why I am able to achieve a +-0.5V on the output (at least within LTspice)

Finally, and most unimportantly, the light PURPLE fields have a simple ultra high-gain difference amplifiers that will detect if the output current and current set are the same, and turn On or OFF the respective LEDs. The green LEDs are voltage-mode indicators, and the red LEDs are to show when current-limiting mode comes on.

Picture of PSU design (major revisions): Transformer calculations help?
-max- (author) 3 years ago

I have the full LTspice .asc file for this and have specific values for all the resistors, if anyone is interested. The darlington pairs are made up of a MJE3055T and a PN2222 fot the 1st stage. the constant current transistors will probably be a PN2222 and PN2907.