**Brushless motors parametrization.**

Motor data:

Kv= RPM per volt.

Rm= motor resistance in Ohms.

Io= No load current.

From Hendershot and Miller:

**Kq= 30/(pi Kv):** Kq= Torque constant.(units = N m)

RPM and TORQUE at current I:

**RPM= Kv(I – Io)**

**Q= Kq(I – Io)**

MOTOR EFFICIENCY:

η= Mechanical power out/electrical work in

**η****=** **(V- I Rm)(I - Io)/(V I):** V and I= working voltage and current.

CURRENT AT MAX EFFICIENCY:

**Imax= sqrt(V Io/Rm)**

Torque at max efficiency:

**Qmax= Kq (Imax – Io)**

RPM at max eff.

**RPMmax= Kv( V – Imax Rm)**

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** **

CURRENT AT MAX POWER OUTPUT:

**Ip= (V + Rm Io)/(2 Rm)**

Torque at max power output

**Qp= Kq (Ip - Io)**

RPM at max power output.

**RPMp= Kv( V – Ip Rm)**

** **

** **

**MOTOR MAXIMUM EFFICIENCY:** (Better mechanical output to electrical work ratio)

η max= [1- Sqrt ( Io Rm / V )²]

As shown above ………..

Torque at max efficiency:

**Qmax= Kq (Imax – Io)**

RPM at max eff.

**RPMmax= Kv( V – Imax Rm)**

** **

** **

**USING THIS EQUATIONS:**

A device is wanted to be moved through water by means of a propeller arrangement.

Total device drag is known: Rt (Total resistance).

Propeller is to be choosen from:

Intallation requirements, physical constraints (maximum allowed prop diameter, …. Etc)

Thrust >> Rt (N)

Torque: Motor is able to turn the prop at required RPM.

Maximum design speed: Vmax (m/s)

Hence:

Required propeller thrust power: Pp= Rt x Vmax (Watts)

Device acceleration could also be imposed by thrust and Rt comparison. But for such a ROV, its not needed.

Now, let’s state our working parameters:

**Are we going to work at Max power output, or at max motor efficiency ?**

Regarding to battery life, max efficiency is a better option.

THEN:

**Imax= sqrt(V Io/Rm)**

**RPMmax= Kv( V – Imax Rm)**

**Qmax= Kq (Imax – Io)**

** **

From Propeller series curves and/or software …..

Choose propeller (with limited diameter, play with Area and Pitch):

1-] Diameter physical limitation

2-] Propeller required torque at RPMmax. = (approximately) Motor torque output.

3-] Propeller output thrust > Rt.

Please note that required torque must include any torque in the arrangement. (Bearings, seals … etc).

Of course, the “perfect” propeller is almost impossible to be found, but a very good approach can be achieved.

Once the propeller is choosen, motor working parameters can be back-calculated.

Equations above, also allow for finding the best approach to a motor, from known Rt and propeller.

Playing around with the above equations, Power input, power output, RPM for a given prop, or device speed for given current, and many other calculations can be made.

Hope this can be useful for you.

Best regards

Notes:

Increasing pitch, decreases RPM, increases current for a given motor.

Same with area.

The more RPM the more noise, the more cavitation risk.

The deeper the less cavitation.

More area, less RPM, better than less Area, more RPM.

For high underwater RPM: Scimitar propellers.

Low underwater RPM: Kort propellers.

No rake required (No ventilation risk)