Winding EI core Stepup transformer for inverter
I'm going to give you a start on designing the primary inductance and choosing the core to see if you grasp what is going on: -

Conceptually, an inverter might step up the 12V dc to a higher DC voltage then use a H bridge of MOSFETs to generate to a sinusoidal 220V RMS. This can be done (also using a H bridge on the 12V side driving a transformer) at a fixed 50:50 mark-space ratio. This means there are no 50 Hz or 60 Hz waveforms passing thru the transformer and this significantly reduces transformer size.

Because H bridges are on both ends (DC input and rectified output) the transformer step-up turns ratio is simply 220 to 12 = 18.333 to 1. For reasons of providing regulation due to 12V battery droop this might be made to be 20:1 to provide a little bit of overhead. Regulation can be accommodated in the high-voltage MOSFET H bridge typically.

With a primary winding of "several" turns and, with the secondary ignored for the moment, you have to ensure the core doesn't saturate and this requires knowledge of the switching frequency to be used (say 100 kHz). Saturation will happen more easily in no-load situations so it's sensible to attack this first.

If the primary inductance is say 100 uH, the peak current into it is determined by V = L di/dt where dt is one half cycle of 100kHz i.e. 5 us. V is the DC battery voltage driving the primary.

di = dt x V / L = 0.6 amps (peak)

Then choose a core to see if things stack-up. Here is a useful guide from Ferroxcube on page 29 of their data handbook: -



Base your core size on this table and/or other manufacturer's equivalents. Page 29 also gives some recommendations for core materials and usually their 3C90 material is "about right" for applications running less than 200 kHz. 3F3 is a better choice (lower core losses) if it is available in the core size you choose.

Estimate how many primary turns are required to get 100 uH (use the \$A_L\$ figure). Then calculate H (magnetic field strength). H is simply ampere turns of the primary divided by effective core length (figure given by any core manufacturer).

H is important because, given the stated permeability of the core, you can calculate peak flux density and see if there is significant saturation.

"Typically" you might choose a core with an \$A_L\$ figure of 1 uH/turn\$^2\$. This tells you that you need 10 turns to make 100 uH. The core might have an effective length of 100mm so now H = 0.6 x 10/ 0.1 = 60 At/m. The relative permability might be 2000 and so B (flux density) will be: - 

\$4\cdot\pi\cdot 10^{-7}\cdot 2000 \cdot 60\$ = 0.15 teslas

This is nicely below the top limit for where you are aiming for. Ferrite saturates significantly at about 0.4 teslas so, depending on calculated flux density, you might either change the material or, introduce a small air gap or, change the core size. 

Review the core data sheet - they will show the "BH" curve and this will tell you clearly where saturation is becoming significant. 

In this example I'd be tempted to make the core a tad smaller but remember, I'm not working with real numbers - I'm pulling numbers out of thin-air just to give you an idea of the process.

It can be a bit of an iterative process at this stage going up or down a size and maybe deciding on putting a gap in the core. A gap means much less saturation for a given ampere turns and to counter this (to get back to a reasonable primary inductance) you'll need to wind more turns to keep the current (and saturation) down to an acceptable limit. 

There is a benefit - you might need to double the inductance (because of introducing a gap and maybe halving the effective magnetic permeability) but, you'll only need to apply \$\sqrt2\$ more turns to restore the inductance. Ampere turns obviously increases by \$\sqrt2\$ but because permeability has halved, B (flux density) has dropped by \$\sqrt2\$.

On the other hand, more turns means higher I\$^2\$R losses so bear this in mind. There is a lot to consider when adding an air gap - sometimes it can be more beneficial increasing the core size.

Anyway, that's the general sort of idea to design the primary winding. As for choosing the bobbin, once the core is chosen the manufacturer will stipulate options. As for wire size, you have limited space on the bobbin and I'm not about to go into the options here because this "half" answer is too long as it stands!