Here are results for 0716 (17500) brushed motor (Betafpv) loaded with 4 blade (Betafpv), 3 blade (Betafpv) and 2 blade (cut made of 4 blade)

*Below we will use “sensor 0” for V, I sensor (1ms, 0.1 Ohm, ina219) at the power supply; and “sensor 1” for I, V Hall sensor at the motor.*

*In all tests 250mAh LiHV battery (charged before each test) was used*

*Propellers:*

**1) Raw experimental data**

*Fig. 1. 4 blade propeller: Raw data as obtained with 60 second run from 0 to 100%. Vo is a mean value of voltage at supply, v1 is voltage taken at the motor when current >0 (see previous post)
*

**2) The Fit**

Experimental data were fit using the model discussed in the previous post. Online tool is available here.

There are 3 steps.

a) We enter available or measured data: Kv=17500 (manufacturer data); I0=0.05 (no-load current, measured by me); R0=0.66 Ohm (measured by me with 4 point scheme at 100, 200 and 300 mA, remember, motor must be mechanically stopped to avoid back-Emf); V=3.5 (voltage at sensor 1), propeller diameter=0.031 m (as per specs)

This gives us all possible values that can be obtained with a load from zero to the moment when motor stalls.

b) We should find now how much load is created with our propeller. It can be done using any value, I am using RPM, as they are most accurately measured. Turn on “propeller” checkbox, turn on “throttle(PWM)” checkbox. As an X scale I selected “electrical power” (better to use physical values more related to thrust and power). Draw all the experimental values vs (Pe=I1*V1). From the experimental plot e.g. we get RPM=32000 at P=4W. Now we adjust Cp propeller parameter to get similar vales in the theoretical plot. Here is RPM and I, when Cp~0.48

c) Now we should fit measured thrust value to theoretical one by adjusting prop. efficiency parameter. E.g. measured thrust=9.5g at P=4W. We find that this thrust corresponds to 0.33 value. Here is the plot.

All done. Plot all possible y-values against electrical power, export to csv and plot theoretical values together with experimental:

*Fig. 2 Experimental (points) and theoretical (dotted lines) values plotted vs electrical power at the motor.*

Amazing, we got almost perfect fit! And we even did not do any LMS multi point parametric fit. We just found 2 parameters (not parameters of the model, but parameters that describe the propeller; propeller load and propeller efficiency) using 2 experimental points (RPM(4W), Thrust(4W))!

*The model works well and can be used for simulations to predict behavior of unknown motors or propellers working at different battery voltages.*

Some of the curves looks different from what is known for brushless bigger motors. Normally, for brushless motors, efficiency g/W has higher values at low power (throttle) and lower value at maximum, (see for example http://rotorbench.com or https://fishpepper.de ). This is predictable, because g/W is just the derivative of the thrust curve (vs mech. power), which is proportional to Pm^2/3. Here we see almost flat dependence in the whole range, it does mean, that natural motor efficiency (related to coil resistance) compensates (make it lower, or worse) g/W curve at the left side. This correlates with FOM which also looks unusual compared to typical bigger brushless motor.

This is also more-or-less clear. Now we know all the parameters and can plot full picture. This picture (online plot here) corresponds to all possible values at a fixed voltage, and the label indicates where we are with our propeller at maximum throttle:

*Fig.3 Motor curves and a propeller load. Labels indicates points where propeller creates load running at maximum throttle at 3.5V, at this point propeller’s mech.power and torque are equal to the m.power and torque created by the motor.*

We see that our brushed micro tiny whoop motor working far from maximum of motor efficiency (magenta curve). Typical brushless big motors are working near the maximum. In the big motor word we would say the motor is overloaded with the propeller. Unfortunately, we cannot get high efficiency because tiny brushed motors have relatively big resistance of coils: we need enough torque, to get enough mechanical power; that means we need to use many turns of coil to have enough torque which is proportional to the turns, coil height, radius, and field; and simultaneously contradictory requirement to have thin wire to have low weight and a gap to magnets.

Sometimes we can read that brushless coreless motors can have as much as 99% of FOM efficiency. Sure, no problem, but without load, unfortunately this is not our case.

With brushless motor the above can be improved: brushless motors have core with high magnetic permeability, so, to create the same field it will need less turns, so they have lower phase resistance and they do not care about depth of the coil, because magnetic gap is created with the core, not the coil. Drawback is 3phase operation: higher PWM and total resistance=2 Rphase (or 1.5 Rphase if encoder is used). Brushless is another story, I’ll leave it for further discussion when I’ll get reliable data on micro brushless motor.

I also have 4 blade props from Crazepony and Eachine but they look identical to Betafpv props, so I believe they will show exact same data.

**3) ESC efficiency**

Brushed ESC is just a transistor with the motor in its collector/drain path and PWM voltage applied to the base/gate. Active loses are resistance of fully opened transistor and dynamic loses are related to charge/discharge current in the transistor while switching. PWM frequency is relatively low, so we expect dynamic loses are low. I’ve tested with 480 Hz PWM (default) and 2000Hz PWM (set in betaflight) and found no difference.

Measured resistance of opened MOSFET is about 34 mOhm.

There is no noticeable difference between power at the motor and power at supply (after supply current is offset by constant FC current), see FOM plot in Fig.2.

Also, the edges are very fast compared to PWM frequency, so we should not care about inductance of the coil.

*So we conclude that efficiency (FOM) of micro brushed motor ESC is close to 100%.*

**4) Reduced load, is propeller model works?
**

The next experiment is important. How good is our propeller model. It is based on very simple assumption: Propeller’s mechanical power is proportional to the (rotational.speed^3) and coefficient of proportionality depends on how much air is grabbed by the propeller (airfoil, # of blades). Another coefficient is propeller’s efficiency, it should depend on how much power goes inefficient way (turbulence, side flows, etc). Typically larger propellers have higher efficiency.

So, I cut 4 blade propeller to have it 2 blade. Here is the plot, for comparison, 4 blade propeller is also shown.

*Fig. 4 Raw data of 2 blade and 4 blade propeller with 0716 motor.*

Let us see what will happen if we will try a) fit 2 blade data the way we did for 4 blade prop (independently on prior knowledge that 2 blade prop is cut of 4blade) and b) fit the data with the parameters obtained from 4 blade propeller, with the only difference to use Cp coefficient reduced in 2 times.

*Fig.5 Fit of experimental data for 2 blade propeller, dotted lines (better fit) is a fit with Cp=0.27 and prop.eff=0.29; and dashed line is a fit with the 4blade model with Cp/2=0.24.
*

We see that 2 blade propeller can be fitted well enough with all parameters for 4 blade prop with the only assumption that it grabs 2 times less air than 4 blade. Not that perfect if we consider it independently, but very good. We also can see that 2 blade prop has a little bit less propeller efficiency than 4 blade.

5) 3 Blade propeller

Airfoils of BetaFpv 3 blade and 4 blade propellers are different, so we cannot use them to check Cp assumption, but nonetheless they are good indicator of how all values are interconnected.

Here are raw data. To compare them with 2 and 4 blade props, data for 3 blade propellers are show with circles together with the data for 3 and 4 bl propellers.

*Fig. 6 3 blade raw data, 3 blade is shown with circles together with data for 2 and 4 blade props.*

Thrust dependence vs time (throttle) is almost the same to the curve for 4 blade propeller. If someone will use only this thrust dependence (typically the only data available on the net) he may say the propellers are identical in performance.

Let us look at big picture. Here are experimental data (points), best fit of these data (dotted line) and, for comparison, the best fit for 4 bl propeller (which coincide with 4 bl experimental data, see Fig.2) (dashed line)

Fig. 7 Data for 3 blade propeller (vs Pe) (points), the best fit (dotted lines) and the best fit for 4 bl prop (dashed lines).

Thrust curve slope is higher than for 4 blade, so propeller efficiency is higher. There is no contradiction with Fig. 6 where thrust for 3bl and 4 bl looks the same, curve for 4 bl (thrust in Fig.7) lasts longer and reaches maximum at ~5.4W at exact same thrust as 3 bl. Higher slope indicates less current at the same thrust, since we don’t know current if we use only thrust dependence vs throttle in Fig.6, they look identical.

The best fit gives Cp=0.40 and prop.eff=0.4.

3 Blade propeller grabs less air and has slightly better efficiency than 4 blade propeller. From practical point of view they are better than 4 blade props (of these 2 specific props), they produce the same thrust, more efficient and lightweight. Not much, but noticeable.