Hi Folks -
I thought I'd amend this old thread since I had pretty much the exact same problems with an Ebay purchased EC2002. I bought it with known problems, which were that the display would jump up and down by a hundred degrees (F) or so at a time and sometimes black out entirely and then periodically wink in and out. Also the iron would only heat for the first minute or so after power ON, and then cool back down to room temp and not heat up again until the next power On. I pulled it open and did some checking.
The first thing I found is that on the Display board, the -5 volt supply had essentially no DC filtering happening, so on pin 34 of the Harris ICL7136 A/D+Display Driver combo chip instead of a nice smooth -5vdc I got raw, half-wave rectified AC. Adding a 10uF electrolytic on this rail smoothed out the erratic display readings. I didn't bother to remove the original fail cap. I'm hoping it just failed open and I'll never hear from it again. Bad assumption probably, but I'm lazy.
The iron still would only heat once on power-up and then not again. I put a scope on the gate of the triac and watched its behavior during the power on phase. While the iron was heating up right after power up, I'd get a roughly 2v square wave on the triac's gate. Once the thermocouple had reached the set temperature the gate would fall to 0v. As the iron temp fell back below the set temp, you could see the Weller chip try to turn the triac back on but the signal on the gate would be a minuscule 200mV AC or so, insufficient to turn the triac on. Back to square one. Having read some of the previous posts on this thread regarding cap failure on these units, it occurred to me that I had not check any of the other electrolytics. Sure enough, on of the 1µF caps on the display board was failed open. I replaced it and Viola! The problem restarting the triac after a temperature cycle was cured and the iron would hold the temp as set.
Here's a pic of my board...Well, I was going to post a pic, but I could not quite get the forum to accept my attachment. Ah, well...