This Instructable is put together with the goal of getting AC-transformers working with LEDs without diodes and capacitors. I will show enough maths here so the concept is applicable to most other AC-only transformers.
Interestingly, many Black&Decker Dust-Buster transformers are AC only, and they are well suited for conversion, since many only use 1/2 of the output (half-wave rectification) only.
Step 1: Working the numbers
First, we have to be aware that the 10V rating is known as the RMS voltage, and is the effective average power of the sine-wave. The maximum voltage, which we will subject our LEDs to, is about 1.4 times higher.
We can demonstrate this by hooking up our transformer and taking some measurements.
The second image shows 10.8 VAC, which the unloaded output of the transformer. So we should expect a peak voltage of 1.4 x Vrms or 15.3v
Next we add a simple diode with a smoothing capacitor and measure the voltage across it: 14.5VDC.
This number is about .8v less than our calculations because the diode has a voltage-loss across it of .8V
This is one reason we try to avoid diodes because each one inherently loses (as heat) a bit of power - .8v is 25% of the power for a 3.2v LED.
So, we will be using 15.3 volt as the basis our calculations.