How to wire C wire to the transformer of HVAC?
If we read your manual correctly it states to install the PEK.

https://www.ecobee.com/wp-content/uploads/2013/12/ecobee3_InstallationGuide.pdf

You can also look here:
https://www.stevejenkins.com/blog/2014/09/how-to-install-an-ecobee3-smart-thermostat/

And a guide to the wire color codes is found here:

Why are the white and orange wires both connected to my thermostat&#39;s W terminal?

That said if you are looking to get around the PEK - of course I can understand that you see 'No Control Board' or Terminal Connections but simply a bunch of things wire nutted together, that maybe this is why you skipped that item. 

In order to do this - you need 24VAC for that common wire and it should follow the color code for the common wire [BLACK] at least black tape both ends of your blue wire if you can't pull a black wire.. (as this tells people what circuit in the HVAC it is supposed to be) , of course that is only if you are interested in doing it right.

In Picture number 2 You have a Green , White, Red and Black wire coming down from top left.  Now since there is not a real drawing of what is where and how it is hooked up and your pictures are a bit difficult to trace the wires ..

I am guessing that the Black wire is the 24VAC common that you will need to connect to - it appears to be wire nutted in the back to a brown or purple wire - I am guessing that goes to the condenser unit. First check it for voltage with a volt meter on AC setting and measure between that wire and the frame screws of the unit. Do not hold the meter by the metal leads - hold the plastic of the leads and measure - do not touch any metal with your hands! You can get electrocuted or shocked. 

After measuring and making sure it is 24 Volts AC - you can hookup your blue wire as long as you taped a black band around the end with black tape - to identify it as really being a black wire. Now you have the proper voltage and wire.

EXCEPT you already hooked up stuff previously and let the smoke out of something, so depending on how much smoke was let out.. your unit should be able to run.

All of this said : Your system should work if you did not burn up anything previously.