As is stated, the time calculation (Faraday's law of electrolysis) is
<liters of solution> * PPM * 15 / <current in milliamps passing through your cell>
As stated above, the equation is agnostic to electrolyte (presence or absence).
The only relationship to electrolyte is that without an electrolyte, its going to be damn tough to get even ONE milliamp through the cell. Water is a pretty damn good insulator to the flow of electricity.
I fail to see why you'd want to produce ionic silver without using an electrolyte, not the least reason being that the electrolyte is required to "break" the normal action of the electrolysis cell (electroplating cell).
Yes, the electrolysis cell is an electroplating cell where under normal (NO electrolyte) conditions, the silver ions are pulled off the anode, go into solution and are pulled out of solution and deposited on the cathode or whatever is connected to it, thereby electroplating it. This is exactly how an electroplating cell works.
The problem is that without the electrolyte, the cell functions this way so beyond being able to run minuscule current through it, everything you make disappears as silver plating of the cathode.
Yes they use an electrolyte in a real plating cell to increase the conductivity of the water. The electrolyte we use however (sodium carbonate 1M solution) has been chosen to INTERFERE with the uptake of the silver on the cathode - a.k.a. a "broken" plating cell which is exactly what we want as we want to raise the level of silver oxide in water to the desired amount, keeping it all from plating out on the cathode.
You NEED the electrolyte. It has 3 functions, 2 of which are important to you.
1) it lowers the resistance of the cell so you can move larger amounts of current through it so you can achieve your desired result using a stopwatch rather than a calendar (wink)
2) it INTERFERES with the normal plating action of the cell thereby allowing the concentration of ionic silver in solution to go up and up and up though, at room temp (thats 75F and no colder), the absolute limit is about 21-22PPM. Beyond this it will start precipitating out of solution. This means you can never make more than 21-22PMM IS because eventually, even if you're heating the cell, it will cool down to room temp and anything over 21-22PPM will precipitate out of solution and eventually collect as a grey dust on the bottom of the container though at the size of the molecules, this could take days to weeks to be visible.
3) for making Colloidal Silver where you then need to reduce the manufactured IS to Colloidal Silver, reducing agents only work in an alkaline environment. The amount of 1M reducer stated (1ml/liter - a.k.a. 20 drops from a standard eyedropper per liter of water) raises the PH of the water to around 8.5PH which is kind of the sweet spot for the reducers we use to function properly and quickly.
And just so its said, Faraday's law of electrolysis ONLY works if the current through the cell remains CONSTANT over the interval you're timing.
What the silvertron does is for each sampling interval (1/100th of a second), it measures the current, calculates how much silver has been pulled into solution during this time based on the constant current and maintains a running total. When it gets to the PPM you set it to, it ends the run. This is not something you can easily do by eye. In this regard, the time interval is 1/100th of a second where the silvertron does it over and over and over until the desired PPM is reached.
This is why Kephra says that the Silvertron isn't overly concerned with the stability of the constant current. Yes it has a constant current limiter but so long as the current over each 1/100th of a second interval remains constant (in a Colloidal Silver cell, changes don't happen nearly this quickly) it will still get you to the end result (PPM) you desire.
In mathematical terms, what the silvertron does is integrate current over time where the step size is 1/100th of a second.