Hello, I'm TV Tech1 and I'll be working with you to identify and help you resolve your problem.
You gave us a Sony, kdl-46w5710 and then a Samsung model number LE40656A1FXXU.
Samsung had a capacitor issue several years ago due to faulty electrolyte from a specific manufacturer, which Samsung no longer purchases capacitors from. I'm pretty sure those units were built in 2007 or 2008.
Are you working on a Sony or a Samsung TV, and is your power supply this one here --
Do you have any references to this known fault? As far as I'm aware, the only issue with bad capacitors has been that model of power supply back in 2007/2008. I haven't seen Sony ever post any issue on caps, and Samsung is the only one that issued any bulletin regarding the 2007/2008 model.
So to be honest, that reference you stated fits your model number is XXXXX as is the TV brand and model.
Ok, this picture here -- the two pictures on the left side with the capacitors boxed in red, I have seen those capacitors with perfectly flat tops on them, and replaced them anyways. It seems to fix the power supply issues even when the power supply relays don't click repeatedly. The key to those caps are they are usually 10 volt capacitors and either 1000uf or 2200uf (microfarad) capacitors. If it was purchased in 2008, I believe you should replace those four caps with new ones.
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So, I recommend you replace them first and foremost, then see if your TV set restores normal operation.
If it doesn't, I suggest saving this page in your web browser, and contacting me again, and let me know the behavior of your TV set, whether it's now clicking or staying dead. If it's dead, you see the plug nearest those caps that has the extra wide double ribbon wiring? On that cable you have not only the Standby 5 Volt supply (those capacitors) wire, but you have a PS-ON and a BL-ON line. PS-ON turns on the rest of the power supply voltages, followed by the BL-ON which is the Backlight on command line that turns on the power supply's backlight section.
That standby 5 volts feeds the TV's microprocessor on the main board, then it feeds back the PS-On and BL-on commands to the power supply board. All 3 are needed for the LCD panel to light, and then power up the picture and sound circuits on the main board.
If you need more details, I have example pictures of those lines, but on another model. They're pretty consistent across all Samsung LCD TV sets.
As for the capacitors, you can get them online, or even from a Hobby store like Tandy's or Radio Shack. Both of those companies originated in the UK and sold parts for electronic hobbyist's back in the 60's and 70's. We have Radio Shack in practically every small town along with larger cities, but I honestly don't know if they exist anymore in the UK. I've asked about that, but never received any feedback about their existence.