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From website: Notes:

For the transformer, a ferrite toroid (doughnut) works quite well and reduces RFI production. I used one roughly 3/4" OD (30mm). An E core with a bobbin works well also and is easier to wind. On the toroid, a turn is counted every time the wire passes through the hole. On the E core, it is counted each time it goes down through one leg of the E and back up through the other. Note how the feedback winding is wound in the opposite direction than the others. It can also be wound in the same direction and have its connections reversed. If the circuit fails to oscillate, try reversing the feedback winding.

... You wind your own, bro. You're wanting to build an oscillator. ;-)
This image?


This one uses a DPDT switch, which you don't have to include.

If you don't want the switch, just wire each speaker to its own jack. One speaker will be left, and one is right. Be sure impedances match the amp ouput specs.
Depends on the output impedances of the devices you're connecting. If you want to do that with power devices, I advise against it. You could do it to low power circuits with some resistors, and I'd probably use capacitors, too.

Please read,

http://www.tkk.fi/Misc/Electronics/circuits/linesum.html
That's right. If you lose the start wire, you're screwed. Pickup wire is very small, even for coils. We make custom coils at my workplace that are used for magnetic measuring equipment. Nothing we wind uses wire that small. Thin wire really sux to work with. I had to repair a humbucker one time. Luckily, it was the finish wire that was broken, and not the start.

There are a few things you can do if the start wire is missing. One is, throw it out. Or, completely unwind the copper from the bobbin, and start over. Or, if there's a tiny bit of start wire sticking out, you could unwind a few layers and pray there'll be enough to solder to. Doing this will decrease impedance and change the sound slightly.

As far as which wire goes to ground, you'd have to either find the specs for the coils that list that, or check the first coil. The second coil's plus should be the same wire (start or finish) of the first coil, and connected to the first coil's "ground".
^That sux! I despise sh!itty service! Thanks for the info.
ROFL Standard push-on terminals, they are. Try using pliers. White is positive. See the cute little "+" symbol stamped into the black terminal bracket?
Like electrical wires. Solid or stranded. They don't list the gauge, but they look to fit a maximum of 14 or 12AWG. Their purpose is to more easily make connections between PC boards or other circuits.

It doesn't HAVE to be a PC board. You can solder them to anything you like. Why do you want to use them?
Probably won't be a "subwoofer" exactly, but should work for low frequency reinforcement. Turn the horns all the way down and try it.
I think you're backwards with the conductivity thing:

http://www.seed.slb.com/qa2/FAQView.cfm?ID=1186

http://en.wikipedia.org/wiki/Electrical_conductivity

The wire used to connect the electronics inside a guitar doesn't matter very much due to the short runs. Using shielded cable, and better shielding overall, is better, though.
I don't have personal experience, but this place looks promising:
http://www.torresengineering.com/maktubam.html
I don't think anybody should be working on electronics if they don't at least have a multimeter that they can connect to their circuit to determine for sure what they're dealing with. They aren't that expensive, you know. Any auto shop has them. Learn how to use it. You'll come to depend on it.
If the load you add has a high enough impedance, you might not even hear it.
You could connect the output of your guitar through a 1M or so resistor directly to the microphone input of that kit, probably. You'd need to remove the microphone, and connect wires from the guitar output jack.
The power off, standby on method works. You could also use a 1 Meg 1/2 watt resistor. It will take longer, but not that bad.
The 555 timer circuit is the winner.
The problem is probably with both the speaker and the power amp. You could try changing the speaker, but that may or may not improve the sound. If you like the modeling, a better solution might be to acquire a tube PA amp and run the spider headphone output through the PA amp. The best solution would probably be to buy a decent guitar amp and get a separate modeling pedal.
Tremol-no, all the way. A bit expensive for what it is, but unless you have access to a machine shop, it's your best bet.

http://www.tremol-no.com/wheretobuy.asp
http://www.allparts.com/store/bridge-parts-miscellaneous-bridge-parts-bp-2005-010,Product.asp
To me, it's counterintuitive. If anything is modified with something that improves it in any way, then the value should increase at least by the price of the mod, minus the price of the replaced part(s). The same goes for any vintage equipment. If the vintage version is intrinsically better, then it should be worth more. The only reason something vintage could be worth more by it being original, but not being of better quality in any other way, is the "irreplaceable" factor. Otherwise, improving with up-to-date technology should cause the value to increase.
It'll be $50 or more for a new speaker. If you really want to keep your amp and replace the speaker, try parts express http://www.partsexpress.com/webpage.cfm?webpage_id=3&CAT_ID=49&ObjectGroup_ID=584
The amount of power a speaker can handle is determined by the gauge of voice coil wire, number of turns, diameter of voice coil, flux density of the pole magnet(s), excursion and compliance of the motor (cone and spider) etc, etc. You will not be able to calculate the rating. Use it in your application and crank the volume until it begins to sound like sh!t. Then, back it off a bit. There's the rating.
You need an amplifier that drives LEDs instead of speakers. Since you don't know a lot about electronics, and I'm lazy, here's a kit you might be able to retrofit:
http://www.allspectrum.com/store/product_info.php?products_id=1240
The Kustom Dart is an ok platform to modify. It sounds ok being driven with a standard 6BQ5. Looks to me like the circuit is powered by 12VDC. This would only be dangerous to you if your skin resistance is less than 400 ohms. Go for it and see (hear) what you get. The speaker isn't the greatest, but it's not the worst, either. I like the construction and finish of the cabinet. I've played with a couple of these amps and don't hate them. They sound pretty good being driven by a tube output stage. There isn't much difference between SS and tube preamplification. Both can sound good, both can sound terrible. The real difference is in the implementation of the circuits. Experimenting with low voltage circuits is relatively safe. If you decide to experiment with standard tube-circuit voltages, I'd be careful.
The pins get soldered into a PC board, and it looks like there a screw terminals to connect wires to each terminal.
^True. If you're just switching amp channels, the box doesn't need to be shielded. Gorgeous wood would be a nice touch.
It's just like building anything else. Is building a house hard and expensive? How about a car? Kind of overkill, I know, but you get the picture. Find out everything you can about building a tube amp before deciding. The 'net is chock full of information. Do some searching and reading, and I think you'll find your answer.

Here's a good place to start that's been cited probably 1000 times on this forum:

http://www.ax84.com/
When you turn down the volume knob, you're increasing the resistance between the signal out and pickup(s), which degrades the tone. Adding a cap across this resistance essentially bypasses it, as far as the signal is concerned, retaining the tone.
If you're using two speakers on a single output, each speaker should be rated for no less than half of the output power.

If there are two channels each rated for 75 watts, try to keep the speaker wattage rating at or above your amp's rating.

Also, you want to be sure to match impedances (ohms). For example, if a single output impedance is 8 ohms, you'll want either two 16 ohm speakers wired in parallel, or two 4 ohm speakers wired in series. If you have two outputs rated at 8 ohms each, then match each output with it's own 8 ohm speaker.
Try to keep the speaker power equal to or greater than the amplifier power. Since you like harder rock, I suggest these:

http://www.partsexpress.com/pe/showdetl.cfm?&Partnumber=290-490

http://www.partsexpress.com/pe/showdetl.cfm?&Partnumber=290-810
You could get by without a preamp if you used a buffer to match the impedance. But then again, the buffer could be considered a "preamp," even with a gain of 1. Still, output amplifiers are meant to be DRIVEN, not trickled upon. Also, as SYK mentioned, increasing a signal requires amplification. Usually, the preamp is designated for this task, and the power amp's task is to drive a load.

And furthermore... active pickups in and of themselves are preamps.
I think it would depend on the material used, mainly the density. I read somewhere that denser materials resonate less, which reduces the parasitic loss from the string vibration. This causes the strings to vibrate longer, which increases sustain. Why not try it and let us know how it sounds?
I didn't find any schematics, but I'm sure it uses the same components as most other pedals. Read up about the circuits in the pedal building thread. https://www.ultimate-guitar.com/forum/showthread.php?t=630592&highlight=pedal+building+thread
A preamp takes your tiny little signal and makes it big enough to drive the output section. Tone-shaping circuitry, among other effects like reverb and tremolo, are implemented in the preamp section. You could run the guitar signal directly into a power amp, but it won't drive the output hard enough to hear much.

There are lots of differences that have been discussed in many threads on this forum. Try doing a search. Fundamentally, the location of a preamp doesn't matter.
Definitely replace the clipping diodes with germaniums. Does this amp even have transistors? I'm picturing a couple op amps and a TDA7240 or something.