Size the battery from the load and the losses.

Not the 12 hours the nameplate suggests. A 100 Ah 12 V battery holds 1,200 Wh nominally, but at 80% depth of discharge and 95% inverter efficiency only about 912 Wh is usable — around 9.1 hours at 100 W. On lead-acid limited to 50% depth of discharge it is closer to 5.7 hours.

Because deep discharge destroys cycle life. A lead-acid battery routinely taken below 50% may last only a couple of hundred cycles instead of a thousand or more. LiFePO4 tolerates 80–90% comfortably, which is a large part of why it is worth its higher price. On top of that, converting DC to AC costs another 5–15% in the inverter.

Watt-hours measure energy; amp-hours measure charge and are meaningless without a voltage. A 100 Ah battery at 12 V holds 1,200 Wh, while 100 Ah at 48 V holds 4,800 Wh — four times the energy from the same amp-hour figure. Always compare batteries in watt-hours.

Divide capacity by draw: 3,000 mAh at 150 mA is 20 hours in ideal conditions, realistically about 17 after inefficiency. But most devices do not draw a constant current — anything that sleeps needs an average worked out from its duty cycle, and getting that wrong is why battery-life estimates for small devices are so often out by an order of magnitude.

Divide the load in watts by the voltage to get current, then add about 30%. A 30 W load at 12 V draws 2.5 A, so you want a 12 V supply rated at 3.25 A or more. Running a supply continuously at its rated maximum is how it ends up hot and short-lived, especially with inexpensive units rated for peak rather than sustained output.

Yes, and it is usually a good idea. A device draws only the current it needs, so a 12 V 5 A supply on a 12 V 2 A device simply runs cooler and lasts longer. Voltage is the parameter that must match exactly — and so must the polarity, which is a far more common cause of destroyed equipment than the current rating.

The total impedance divides by the number of speakers, so two 8 Ω speakers in parallel present 4 Ω and four present 2 Ω. That draws more current from the amplifier, and once you go below its minimum rated impedance the amplifier either trips into protection or fails outright. Check the amp’s minimum rating before wiring anything.

Parallel gives each speaker the full voltage and more power, but lowers the impedance the amplifier sees, which is the risky direction. Series raises the impedance, which is safe but delivers less power and lets the speakers interact through their back-EMF. If neither gives a load your amplifier likes, use another channel rather than compromising.

By the usual convention, 1 ms drives it to one end of its travel, 1.5 ms centres it and 2 ms drives it to the other end, mapped linearly across the rated range. It is only a convention though: many servos actually accept 0.5 ms to 2.5 ms, and pushing one past its true mechanical limit makes it buzz, draw heavy current and eventually strip its gears.

360 divided by the step angle, multiplied by the microstepping factor. The common 1.8° motor gives 200 full steps per revolution, which becomes 3,200 at 1/16 microstepping. The extra resolution buys smoothness and quiet rather than genuine accuracy, since the holding torque of a single microstep is very small.

Not meaningfully. It makes motion smoother and much quieter, which is why 3D printers use it, but the torque holding any individual microstep is a fraction of a full step, so the rotor can be pushed off the commanded position without the controller noticing. Positional accuracy still comes down to the motor’s inherent step error and the mechanics around it.

XL = 2πfL, with L in henries, and XC = 1 ÷ (2πfC), with C in farads. Inductive reactance rises with frequency and capacitive reactance falls, which is the whole basis of filtering. They are equal at the resonant frequency, f = 1 ÷ (2π√(LC)), where they cancel and the circuit becomes purely resistive.

Filament plus electricity plus wear. A 50 g part at $20 per kilo is $1.00 of filament; four hours at 150 W and $0.17 per kWh is about $0.10 of electricity; add roughly 10% for machine wear and the running cost is around $1.21. What that leaves out is the part that actually costs money: your time, failed prints and post-processing.

Enough to absorb the print that fails at hour nine, which a 0% margin does not. Material and power are the small part of the real cost — labour, design time, support removal, sanding and the occasional total loss dominate. Pricing by machine hour as well as by gram is the usual way to make a light but slow part pay for itself.