http://www.eserviceinfo.com/index.php?what=search2&searchstring=KV-35S35+ch.AA-2D
http://www.givemefile.net/smanuals/tv/sony/file/8230.html
Above are two service manuals, download BOTH and use them to sort out your problem if you can.
The Power supply, always start at that... Anyway please use this Service manual above, inside are all the troubleshooting and repair guides, circuit diagrams instructions the lot.
&, well I applaud you for wanting to have a go and try to repair you set. But PLEASE, start in the right place, otherwise you will simply waste your time and money, You see, you MUST start, in the right place, The power supply, FIRST, as, if, that isn't right, nothing else can be. But you simply cannot do anything without a Service Manual. In this it has everything one needs to repair the set. OK if it is shutting down, after that amount of time, of working and then faulting, then it is most definitely a "Thermal" problem, the way we "Find" problems such as this is "Freeze Spray" you use this on each component that is suspected of being the "Culprit" and you will soon,find that errant component. Now, also not every fault is a Capacitor, I see many, many people that think, that Capacitors are always the "Culprit" and replace them, because someone else had done so and it worked, well that maybe sometimes true, but most Symptoms, can, and do have many, many different "Causes" & also, now while that may be true in a few cases, a faulty Electrolytic Cap, or even, ordinary, Cap goes faulty, this is only a SYMPTOM, something has damaged those caps causing them to "Fail". One MUST always find the root "Cause", otherwise if you just treat, the SYMPTOM, the "Cause" will just make the "Symptom" come back, in short order, as the prime cause still, hasn't been fixed., Now Capacitors hate AC ripple, or Over Voltage. Thats it. Oh.. and Heat, if these Capacitors, are too close to a Heat Source, then thats a prime cause of failure. Resistors, hardly ever go faulty,and if they do, it is generally obvious. However it doesn't hurt to check values with our Multimeter. Remember though resistors are made, with up to 20% tolerance from stated value, as such, are not too critical, unless in Timing circuits etc. Always check, the Values, of resistors, as with any other component, with the power OFF. Now Diodes especially "Zener Diode", are another thing to check, those and ordinary "Signal Diode" should always be suspect. next we have Transistors and IC's. The Transistors MUST ALL be checked to see if they have a good, "Junction" this is done, with our meter set to "Diode Test" and usually are about 0.6 to 0.7. With IC's you must check, voltage levels, or Logic Levels, going in and out, have a look at the circuit diagram, and it will tell you what they should be. It is a good idea to always measure the Power Supply voltages, see if a "Rail" is Low, that will be because some component has gone somewhat or all short circuit, to Earth, now, if the "Rail" is Higher, suspect an "Open Circuit" component, like a Diode, or Transistor. The voltages expressed in the manual are spot on, ANY variation MUST be investigated. I have been doing all this for over 30 years now, and I do indeed wish, we could have a set of "Symptoms" and be ables to say.. "Oh that;s the so-in-so and replace this" however unfortunately this although does happen, most "Symptoms" can have literally tens, of "Causes" all often "Interlinked" One simply has to do a methodical troubleshooting procedure, and always think, "are my Voltages Correct" as this is how, you Fix the problem, by making them so. Keep up the good work. If that link above doesn't furnish a Service manual, it is imperative you get one, even if you must pay for it. It will pay for itself in about 2 mins.
it is nearly always my professional opinion that one should always get a "Quote" from the manufacturers recommended service center, SONY, in your case, contact a local or head office, and ask for a referral to an Authorised service center, I cannot stress how important that last point is. It is even worth the cost of shipping. you see an authorised service agent, will have trained staff, trained on their brand, also they have first hand access to parts, original parts. This will actually save you money and ensure you can actually get it back fixed. Of course asking for a "Quote" also gives on the option of repair or replace. Often my clients,do not go through with repair, unless it is moderate, but in some of the more costly jobs they see it, as do i, that a replacement, with a warranty, is a better option. As an aside here, one could then tear the TV down, and sell the Known Good bits. You'd be surprised who needs parts, i myself often must by from third party sources to effect a repair. So money could be recouped, somewhat,and help out someone else.
