Efficiency

Efficiency & Waste Heat

Efficiency is a dimensionless parameter expressed in percent.  It is the ratio of power output divided by power input.

Inefficiency in any system produces waste heat.

The majority of internal combustion engines convert only about 20 to 30 percent of their fuel's heat energy into useful rotational power at the end of the crankshaft.  The lion's share (70 - 80%) of the original chemical energy is waste heat which leaves via the exhaust and cooling systems.

Electric vehicles have several sources of waste heat, described below. 

Charger Efficiency

Battery chargers for electric vehicles are typically switched-mode power supplies.  They convert AC power from the utility into DC power to charge the battery.  I have measured the efficiency of the 5.7's charger at 87% and the ePure's charger at 93%.

Cell Efficiency

The round-trip efficiency (energy in versus energy out) of a single lithium-ion cell is about 95 - 96%.  For comparison, this number for lead-acid cells is only about 75%.

Battery Efficiency

However, when a number of li-ion cells are connected in series, the efficiency gets a little worse.  The individual cells that make up a battery pack are not all identical.  There may be slight differences in internal resistance, actual capacity, and rate of self-discharge.  A BMS is necessary to compensate for these differences. 

BMS Efficiency

By design, the BMS wastes energy.

The BMS attempts to bring all the individual cell voltages to within about 10 mv of each other.  It does this by resistively discharging the higher-voltage cells to the level of the lower-voltage cells (although there are safety protocols in place to prevent overdoing this and making the battery useless).

More advanced BMS technology uses active circuitry to remove charge from the higher-voltage cells and redistribute it to the lower-voltage cells.   Although much more efficient than a simple passive discharge, the active process is not itself without losses.

Controller Efficiency

The motor controller itself is highly efficient (on the order of 95 - 98%).  This becomes self-evident when you consider the thermal aspects of the problem.  Take a motor operating continuously at 5 kW.  A 2% loss in the controller implies it must dissipate 100 W internally.  The heat generated is thus comparable to that of an incandescent light bulb of the same wattage.

So, to a first approximation, the output power from the controller must equal the input power to the controller.

Motor Efficiency

Motor efficiency is  the ratio of mechanical power output divided by the electrical power input.  Larger electric motor are more efficient than smaller ones.  Very generally, electric motors for small vehicles have peak efficiencies in the 90-ish percent range.  However, efficiency is not a single number and can vary greatly throughout the motor's operating range.  (See the ePure's Motor section for a typical efficiency map.)

A motor's loss terms are copper windings (joule heating), iron core (eddy current), and viscous force (cogging torque, bearing friction, and windage).  

To a first approximation, a motor's no-load power consumption accounts for the viscous force and iron losses.  Copper losses are a function of current squared.  Thus:

Efficiency = (V * I - I² * Rw - V * I0) / (V * I)

Where:

V = Input Voltage

I = Input Current

Rw = Winding Resistance (for a pair of phase windings, as measured by an ohmmeter)

I0 = No-Load Current

So, (V * I) is the power input, (I² * Rw) is the copper loss, and (V * I0  ) approximates the viscous force and iron losses.

The resistance of copper increases with temperature by approximately 0.4% per degree C.  Thus, the hotter the motor gets, the more heat it makes in creating the same torque.

Efficiency is zero at no load because no mechanical output power is being produced (all the input power is being used just to overcome internal motor losses).

By inference, efficiency is poorer at light loads because the no-load losses are still relatively large compared to the input power.

Similarly, as the load increases, I² * R losses increase exponentially and efficiency is again reduced. 

At elevated temperatures, the torque constant Kt (specified at 25 degrees C) degrades because magnetic flux decreases.  Old-tech ferrite magnets fall off at a rate of about 0.2% per degree C.   Neodymium magnets are much better at 0.11% per degree C.  Samarium-cobalt can be as low as 0.025% per degree C.  And the good-old Alnico (aluminum, nickel, cobalt) magnet only loses about 0.01% per degree C.

Specific Energy Comparison

Below is a quick calculation to illustrate how a battery compares with a liquid fuel on an energy-per-mass basis.  The following numbers are representative at the time of writing (December 2023) but will undoubtedly change.  My point is to illustrate how far batteries must go to catch up with liquid fuels in terms of specific energy.  This is important for minimizing the weight of the vehicle while maintaining a useful range (which would be imperative for an application like flight).

Currently, a very high-performance battery chemistry is lithium-sulfur.  It has a specific energy of around 450 Wh/kg.  Note that this is roughly double that of the present lithium-ion technology.

Gasoline has a specific energy of 46 MJ/kg.  A joule is equivalent to a watt-second, so we need to divide by 3600 to compare with watt-hours.  The result is 12700 Wh/kg.   That's a factor of 28 times greater in gasoline's favor.

The picture looks a little better when we also consider the efficiency of the prime mover.  An internal combustion engine is only about 30% efficient under optimal conditions.  So, 12700 Wh/kg * 30% = 3810 Wh/kg (effective).

Whereas an electric motor/controller might be 90% efficient under optimal conditions.  So, 450 Wh/kg * 90% = 405 Wh/kg (effective).  Still, that's an order of magnitude worse than gasoline and an internal combustion engine.

Of course, once you've burned the gasoline, that's it.  You can refill the battery for pennies per kWh over a life of 500 to 1000 cycles.  One final thing to consider is that a battery's specific energy diminishes as its number of charge cycles increases.  It may yield only 75 - 80% of its initial capacity after 500 charge cycles.

P.S. My wife suggested I mention for comparison that hydrogen has the highest specific energy of any practical fuel at 142 MJ/kg (39400 Wh/kg).