Well, I dumped a lot of info in my last post, but didn't really draw any conclusions. As far as new cells vs. old ones, I still think the harvested cells are the way I want to go. This is mainly because I'm harvesting cells anyway for multiple projects, and I'v got the time to process and build the cells and batteries. If your purpose is singularly to build an eBike, then building one from new cells, or saving up for an off-the-shelf unit makes more sense.

Anyway, I thought in this post I'd share my battery calculations. I figure if they are correct, someone might benefit from them in the future. If they're flawed, maybe someone will correct me. I know almost nothing of calculating range or miles/Wh or torque vs. speed or uphill power. I've read the wiki, and played with the simulator on ebike.ca. I'm afraid its all still over my head. But I do know cells and batteries, so I'll share that.

To determine my battery requirements, I have to know my power requirements. That is, I need to know what the motor and controller need in order to function. Some of these things I know, and some I've had to guess at due to lack of experience, but Google and YouTube have helped a little.

Here's my eBike's power requirements:

Voltage: 48V. I figure the higher Volts means fewer Amps per Watt.

Watts: 500W. Another guess. I think this will be sufficient for what I need.

Hours of operation: I'm hoping for at least 1 hour of bike riding time.

Amps, Ah, Wh: These will be calculated values based on Volts, Watts, and Hours.

Another assumption I've had to make is how fast I want to go. I figure 5mph uphill, 10mph on flats, and 15mph down hill. Yep; just call me Captain Slow. Also, I intend to pedal through the ups and flats. The motor is only there to assist, not to drive. This is why I'm thinking 500W will be sufficient.

Knowing the Voltage "category" is great, but I need to know the actual range of Voltage I have to work with. For that, I'll need the first few characteristics of my battery and cells. I'll be using 18650 Lithium Ion cells which have a voltage range of 2.8-4.2V, 3.7V nominal.

For a 48V system, I can choose 13s or 14s for how many cells will be in series. I chose 14s. My battery voltage range will be 42-58.8V, 51.8V nominal. Just multiply everything by 14. For safety margin, I'm choosing to limit the battery discharge to 3V per cell rather than 2.8V. This will reduce the useful capacity somewhat, but you'll see later, this really doesn't matter much.

Now for Amps. I can go about this a couple ways. First is to see what the Amps will be assuming constant 500W through the range of volts from the battery, full to empty. So with W / V = A, I can calculate the range of Amps needed as 8.5A (full) to 11.9A (empty). This averages out to roughly 10A.

Another way to look at it is to limit the Amps and see what the Watt range will be. In this case, at 9.5A I'll have 558.6W (full) to 399W (empty). Roughly an average of 480W.

For now, lets say 500W, and with a nominal voltage of 51.8V the Amps will be 9.65A.

Ah, and Wh is easy. Just multiply by hours of operation (1). Or Wh can be found by V * Ah. Same result either way. Now I have the full list of requirements for the eBike:

Next I need to know how many cells to put in parallel. This is a little tricky, as there are two variables which have to be within a certain value range. The first is Amps. Each cell is going to have a maximum safe Amp draw. In the case of old laptop cells, this is usually somewhere between 0.5-1A. The more cells in parallel, the less current is drawn from each cell under a given load. In this case, I'm going to go for a current limit per cell of 0.55A In the past, I've pushed these cells as far as 3.8A each, but only for a few seconds at a time. With the eBike the draw will be constant, so I want to be pretty strict about the Amp limit if I can.

With the battery current being around 9.65A (assuming full-throttle and nominal voltage) I'll need 9.65A / 0.55A/Cell = 17.55 cells. Of course, you can't have part of a cell, so round up to 18 cells in parallel. The actual current draw per cell can be calculated with 18 cells (parallel) * 9.65A = 0.536A/cell. So that's nice.

The other requirement for determining cells in parallel is the overall Watt-hours. This is important because it will determine how long the battery will actually last between charges. Each cell Wh can be calculated: 2Ah * 3.7V (nominal) = 7.4Wh. The Wh of each cell will add up to the total Wh of the battery. So the required 500Wh (battery) / 7.4Wh (cell) = 68 cells, minimum. But these need to be arranged in a 14s configuration, so this will need to be taken into account. 68 cells / 14s = 4.86p. Again, you can't have part of a cell so this is rounded to 5p.

Only 5 cells are needed in parallel to meet the power requirement of 500Wh, but 18 are needed in parallel to meet the current requirement without overdrawing each cell. So the larger of the two numbers is chosen, and the result is that a 14s18p battery (252 cells) is needed to run the eBike. Clear as mud? Here's the whole thing laid out on my spreadsheet:

But wait, there's more!

Now that the battery configuration is know, we can move from estimated values to actual ones. Some of these were known already, but some are new properties that can be found as well.

The Voltage was already figured: 14s = 42V to 58.8V, 51.8V nominal.

The battery can be safely operated at roughly 10A. This was also already known.

Capacity though is now able to be calculated: 18p * 2Ah/cell = 36Ah. (And I was hoping for just 10!)

Watt-hours can also be calculated: 51.8V (nominal) * 36Ah = 1864.8Wh.

Hours between charges can be calculated too: 1864.8Wh / 500w = 3.73h (assuming full throttle the whole time)

A single 14s18p battery is going to be big, heavy, and unwieldy, so I've elected to break it up into 6 smaller batteries, 7s6p. So these will be the properties of the whole battery bank. I've already built 2 of them, and the other 4 are assembled, but still need to be soldered.

Another fun bit of info that can be found is the battery's overall internal resistance. I can't do the whole thing yet, but I can do the IR of each smaller battery as I complete it. For now particular reason, I soldered batteries 3, and 4 first. So I'll start with battery 3. IR can be calculated as IR=ΔV/ΔA when it is put under load. Batt3 has a resting voltage of 26.4 at the moment, and the largest resistor I have is a 50W which measures in at 10.4 Ohms. Putting the resistor across the battery, the Volts drop to 25.4V. Change in Voltage is therefore, 1V exactly. The current running through the circuit can be calculated: 25.4V (load) / 10.4 Ohms = 2.44A. Then, 1V / 2.44A = 0.409 Ohms. This will likely be typical of all the batteries.

Here's where I'm not sure about my math...

I figure if each battery has about the same internal resistance, I can estimate the total IR of all six put together. Basically, 6 resistors, 2 in series x 3 in parallel. Series is easy: add each value. I'll change 0.409 Ohms to 409 mOhms to make it easier. Each series will be 818 mOhms. For parallel, 1/R1 + 1/R2 + 1/R3 = 1/Rt. Then, 1/818 + 1/818 + 1/818 = 0.00367. Then 1/0.00367 = 273 mOhms.

I'll have to come up with a 10 Ohm resistor that can handle 260W or more to check that in real life.

Speaking of real life, what do all these values mean for my eBike? No idea! LOL

I hope it means that the wheel will turn when I twist the throttle...