Any core without a gap will have its maximum value of permeability. Adding a small gap can significantly reduce the permeability of the gapped-core and this reduces peak flux density significantly for the same number of turns and current flowing.
It's all contained in this formula, B = \$H\cdot\mu\$
where B is flux density, H is magnetic field strength and mu is the actual magnetic permeability of the core (or core with gap).
H is the ampere-turns of the excitation coil (primary) divided by distance around the core. If you add a gap that reduces B by 2 then, you should add turns back-on to restore the inductance of the primary. The beauty is that inductance is related to turns squared so if B halves (due to the gap) then you need to increase turns by 1.414 to restore the inductance.
But, this increases H by 1.414 so the halved value of B increases by 1.414 to 70.7% of where it was originally.
So now you have a transformer with a gap that has exactly the same inductance as before but only 70.7% of the peak flux density (at the expense of more turns). If you can fit the extra turns on and reducing B to 70% is acceptable then you have a simple solution.
Of course you'll now have bigger copper losses and you might be fighting a battle that cannot be cheaply won. Keeping the turns as low as you can is a big help - having no more turns than necessary is the trick.
First I'd suggest you to double check your power requirements. You are consuming 12+2.5 = 14.5 watts of power which seems too much for a home automation product. If you are just running some relays, micro-controllers and sensors on that power, you are probably doing it wrong. Again, you might be running some extra load on it and it might be correct.
Now coming to your hardware selection, there are a few questions that you need to answer first:
1) Are there any physical constraints on size of the power supply?
2) Are you going for a bulk production?
3) Do you have some knowledge about electronics and are you willing to put some time in R&D?
If your answers are mostly "NO", then simplest thing would be to go for a transformer based power supply. You can choose a center tap transformer or two different transformers - One for 12V and other one for 5V. Add two linear regulators 7812 and 7805 to make things more robust and you are done. Obvious cons will be high cost and bigger physical size but it will get you up and running very fast. Alternatively you can just buy two power adapters.
However if your answers are mostly "YES"(which is the case actually), then I'd suggest you to go for switching power supplies. Why? Low cost, smaller size and higher efficiency. In the cost of one normal transformer, you will have complete power supply designed if you choose your part vendors wisely. If all major companies are giving you an SMPS based chargers, there must be a good reason to it. If you have decided to go for this, these are the points you need to research on:
1) Selection of an offline switcher IC. You need to choose an IC suitable for your power requirements. For ex - Viper22a is one of the commonly available ICs and it can give you around 20 watts of power if your input voltage is in the range 195V to 265V.
2) Selection/design of the ferrite core transformer - This is the toughest part in my opinion unless you have a good knowledge about magnetics. A cost saving idea would be to go for multiple secondary windings - dual to be precise. One for 12V and the other one for 5V. You need to select a suitable transformer core for the power requirements. You can follow app notes for this purpose.
3) Once these two key components have been selected, you need to pay some attention to noise. What's the max ripple noise you can accept? If you aren't performing any precise adc measurements, 5-10% of output voltage is usually acceptable. For analog stuff, you need rock solid power supply or you will end up introducing some measurement error. Once the acceptable noise has been agreed upon, you need to design a suitable filter for the same. In some cases, just a capacitor is enough. In other cases, an LC filter might be required.
Note: There is yet another cheaper version - Non isolated power supply where you use an inductor rather than a flyback transformer. However you are always at a risk of shock if someone touches the circuit and usually not recommended.
Best Answer
IMPORTANT: If you use a TVS or a MOV, you MUST USE A FUSE IN SERIES WITH THE HOT LINE. This is non optional. MOVs tend to fail shorted, and so can TVS diodes. In the event this occurs, your options are use a fuse, or start a fire. Fuses are the better option.
1. What do these X and Y capacitors even do?
X capacitors, along with their cousins Y-capacitors, are both grouped together and known simply as 'safety capacitors'. In your application, which I assume is class II, you have no earth connection (and thus has a sufficient insulation barrier to qualify as class II).
The purpose of both X and Y safety capacitors is more or less the same. They are both there to reduce the amount of EMI that gets injected into the mains from whatever the power supply is doing. As you know, capacitors become less impeding as frequency increases, so these capacitors serve to effectively short higher frequency noise.
2. What is the big deal about noise?
You always see them present on the input stage of a switch mode power supply, yet they are generally absent on older 60/50Hz transformer-based power supplies. The switching harmonics from the bridge rectifier all get dissipated as tiny eddy currents and hysteresis losses in all that iron and never make it across back into the primary winding.
Switch mode power supplies generally employ square waves (or try to get as close to a square wave as possible) and the harmonic content of that alone is none trivial and a large portion of it can conduct just through the capacitive coupling of the SMPS transformer windings, none the less the ferrite core. Worse, the diodes are directly on the mains, and there is no transformer that could potentially mitigate the diode harmonics.
3. An X across mains is not the only option.
I'm going to assume you have a fully isolated (totally floating) 5V output, as is common for a low power class II power supply.
The problem is nothing is ever completely isolated. There is parasitic capacitance from everything to everything, and everywhere to everywhere. There is probably over a volt of AC potential between any point on your body and earth at this very moment (assuming you're floating and not grounded through something). This is why you get a hum on audio equipment if you touch an input jack with your finger.
Well, the situation is somewhat crappier in the case of your class II power supply. You know that little ferrite transformer? The one with two conductive windings right next to each other? Yeah - they're gonna couple capacitively, but the impedance will be quite high even to high frequency noise. This is going to turn everything on the isolated secondary side into an unintentional radiator. This may or may not be an issue, but one solution is to connect a Y capacitor between the primary ground (the neutral line in your case) and the output ground. If for some reason the polarity of the plug might get reversed, you can connect 2 Y capacitors to the output ground, almost as if it was your Earth.
The idea is to create a much lower impedance path for high frequency noise, shunting it back to the primary side instead of radiating it.
HOWEVER, there is a very important safety consideration here: you've created a path for current to leak across the isolation barrier, and the potential will be potentially dangerous. You must be careful and ensure the Y capacitors are not so large in value to allow dangerous levels of leakage current to flow, because that flow might be going through a person/dog/kitten/whatever.
4. Come on already, how do I pick that X capacitor value?!
It really just comes down to how noisy is your power supply, how much noise from the mains you want the supply to tolerate, and how good of a power factor you want. Power factor is almost always going to be lower priority though, since most countries require you to meet EMI standards above anything else.
A larger X capacitor will survive higher surge transients, will have a more potent effect on differential noise because it is lower impedance to a wider frequency range, and generally just improve things in the differential EMI area. The drawback is that, being across the line, it will constantly be drawing a small amount of apparent power. For example, a 0.47µF X capacitor across the line with 240VAC input at 60Hz (I know 50Hz would be more common, but lets make things worst case) will draw roughly 10W of apparent power at all times. If your SMPS is a 500W ATX power supply, then your power factor is 0.98. Great! If it is a 50W laptop power brick however, your power factor is now 0.8. Not so good. You'll probably want to choose something a little smaller. As the power levels decrease further, it becomes less realistic to achieve a decent power factor, but you also don't really have to care that much. You're building a 7.5W power supply. Let's say it draws 10W of real power under full load. Using a 0.1µF capacitor would result in 2.2W of apparent power draw assuming 240V 60Hz. But that's ok, it's just 2.2W.
The smallest X type capacitor you really see is 10nF, and they can get as big as several µF. I think 0.1µF is a reasonable choice for your application. You'll consume some apparent power, but this is on the order of 400mW in North America. 0.1µF is probably a bit larger than you really need, but with noise, its usually better to have too much noise reduction than not enough. Actually, its always better.
Unfortunately, there just is no real hard and fast rule here though. You can't really calculate the value, because the minimum value is based on the actual conducted EMI and what levels are acceptable for your application. On the other hand, the value is usually not that critical. I usually size it based on a reasonable power factor (but don't get greedy - sure, a power factor of .99 after the input sounds pretty sweet, but it doesn't matter if your brick fails FCC etc.) and, to quote Samuel L. Jackson (in Jurassic Park), I "hold onto my butt" and hope its enough to meet EMI compliance. Thus far, it's not really been an issue, as long as you've kept EMI in mind when designing the remainder of the supply.
Maybe I've just gotten lucky, but it's worked for me so far.
I would ad that a second X capacitor after the choke is the correct placement and will further reduce differential noise and help the common mode choke with common mode noise a little bit too. But it is definitely optional.
5. You didn't ask, but lets talk about MOVs vs. TVS diodes.
You CAN use a bidirectional TVS diode here, but remember, your breakdown voltage will need to be higher than potential sustained overvoltage conditions on mains, and peak voltage is what matters here, so 1.41*250VA = 350V, plus some healthy head room. So lets call it 380V. The thing is, most real surges you might see on the AC line are going to pack some joules behind them, and joules make TVS diodes pop like a bubblewrap scratching post in a room full of angry cats. Sure, you can get pretty beefy ones, but they get pretty expensive ($2+) at the voltages you need. And even then, their clamping voltage is basically the same as a MOV.
MOVs are preferred in the place you've used a TVS. MOVs engage in nanoseconds, and any surge that is powerful enough to make the difference between 10 nanoseconds and 30 picoseconds actually matter is probably too much for a TVS diode to handle anyway. Plus fir 50 cents, you can get a MOV rated for 250VAC line voltage that can absorb almost 200J - that's more energy than the impact of the baseball going 100 miles per hour. Let me reiterate, for 50 cents.
It's better to use a TVS after the diode bridge, but before the actual step-down conversion. Let beefier things soak up the heavy stuff, and then any tiny fast spikes that make it through can be safely handled by the TVS diode.