I know this thread is old, but I just recently ran into this problem. I have an old 19" Insignia LCD (NS-LCD19F) television that is being used in my wife's study at home. This tv has no HDMI, but it does have a component, a composite, a coaxial and a VGA port. I wanted to attach an Apple TV 3 to it, but since there was no HDMI I had to purchase a HDMI to one of those four ports converter. I ended up buying a HDMI to component converter to take advantage of the televisions 720p. I bought a Portta PETHR HDMI to YPbPr Component Converter.
Now this product has a useless manual as it just tells you to attach the HDMI device to the HDMI port of the converter and the television to the component port. It also tells you to plug the power adapter to the converter. There are no switches on this thing. So of course it doesn't work right out of the box. You have to tweak it or actually you have to set up the Apple TV to get this to work. As soon as I attached everything as instructed and set my tv to component, I saw the Apple logo on the screen and then nothing but blue. After doing some research online, I discovered that I had to set up the Apple TV to a resolution that my Insignia tv could handle. So of course the dilemma was I couldn't set up the Apple TV on this tv since I couldn't see the home screen.
I thought all I had to do was to use a television that could handle the Apple TV without the converter and setup the resolution to 720p. I attached the Apple TV to my HDMI tv in the living room and set it up for 720p. When I reattached it to the converter and Insignia tv in the study, all I got was a blue screen again. I tried several resolutions and nothing seemed to work. I was about to return this product.
I then decided to just check if, in fact, this product was defective by attaching it to the component ports of the HDMI tv in the living room. It actually worked! I set up the Apple TV to 720p and then reattached it all to the Insignia tv. This time it worked. So the solution was to not simply set up the Apple TV to 720p, but to make sure when you do it's attached to the converter. So now it works great on the Insignia. My wife is now able to use the Apple TV in her study.