From freelance to CTO: conditions

I am currently working as a Freelance mobile programmer for a start-up since the project started, but I have always given my opinion and advise about business development (for free),
and it came to a point where I think I could get more long-term benefit if I get a % of the company/profits instead of working as a freelance. My idea is to suggest a change in my role, from freelance developer to CTO or something similar.

I would like to know what are the usual compensation terms (A % of the current profit, a % of the company's value, an objective based compensation, a % of the future (possible) profit...? How much?) for a CTO in a startup.

Also, is it required to invest some money to get equal to the founders? When would I start getting a compensation? What happens if I want to quit after a year?

It really depends, are they pre-money? is the company profitable? are they even looking for a CTO? is their product tech-heavy? what size are they? as you know the duties of CTO change dramatically when a company reaches a certain size, from being the lead developer of the project to determine projected costs of new features and hiring other brilliant people.

If they don't have the product yet, no funds and you're their sole developer, you can ask for pretty much whatever and negotiate the terms, a fair amount would be 30-50% for doing all the initial development, vested over time ofcourse.

if they're post money, they can basically hire someone to do this job, so it comes down to what they're willing to pay long-run to secure a good CTO (assuming you're it).