# Motor Quiz 1

**Given:

• 775 Pro motor connected to a fan through a gearbox + belt/pulley

• 80% power efficiency of gearbox + belt/pulley

• 7.8 volts applied to motor

• fan is spinning at 2500 rpm

• 20 watts mechanical power is required at the fan to keep it spinning at 2500 rpm

Solve for the gearbox + belt/pulley gear ratio (from motor to fan)
**
*
*

I calculate ~4.72:1
Spreadsheet doesn’t paste in …

It’s attached.

I used the back EMF constant to compute the motor speed from the applied Voltage minus the I*R voltage drop.

The ratio of the motor speed and the load speed (given) defines a gear ratio.

Given the power at the load, compute the torque on the load side.
Reflect this torque back to the motor side (1/N).

Given the motor side torque, use the motor torque constant to compute the motor current. Here I broke the algebraic loop, by using a separate cell.
The motor current then changes the IR Voltage drop.

This took a few iterations.

… or did I mess something up … Thanks!!

MotorQuiz#1_170127.xlsx (17.7 KB)

MotorQuiz#1_170127.xlsx (17.7 KB)

~22.45 Speed UP and your motor dies 2-3 seconds after starting.

Bonus question! What’s the gear ratio to get the most RPMs/air out of the fan? Hint:Fan power consumption is proportional to the RPM cubed.

I got an error when attempting to load your spreadsheet.

But try checking your answer this way:

With a gear ratio of 4.72, 2500 rpm at the fan would be 11800 rpm at the motor.

Let Sfree be motor free speed at 7.8 volts
Let Tstall be motor stall torque at 7.8 volts

Then the motor torque at 11800 rpm would be Tstall*(1-11800/Sfree)

Knowing the motor torque and speed, compute motor output mechanical watts.

Multiple that by 80% and compare it to 20 watts.

*

For my previous post, I neglected gear/belt efficiency.
Was that the 80% you were referring to?
If it is, I used 100% - not realistic.

I am attaching a revised spreadsheet - I believe I have resolved the error.
I have included the Calculations you provided.
At 100% efficiency, the motor mechanical power and the fan mechanical power are identical.

Along the way, I added an efficiency factor for the gear/belt system.
80% => N = ~4.678, Mechanical motor power is ~25 watts, Electrical motor power is ~30 Watts.

MotorQuiz#1_170127B.xlsx (19.2 KB)

MotorQuiz#1_170127B.xlsx (19.2 KB)

You missed the at 7.8V bit. Separately, I don’t think you can assume power vs speed is a linear relationship…

hmm…

In an ideal motor, voltage vs. free speed is a linear relationship and voltage vs. torque is also linear (therefore voltage vs. mechanical power is quadratic).

Yes:

That looks about right:

``````Maxima 5.27.0 http://maxima.sourceforge.net
using Lisp GNU Common Lisp (GCL) GCL 2.6.8 (a.k.a. GCL)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.

freeSpeed: (motorVolts/specVolts)*freeSpeedSpec\$

stallTorque: (motorVolts/specVolts)*stallTorqueSpec\$

motorSpeed: fanSpeed*gearRatio\$

motorTorque: stallTorque*(1-motorSpeed/freeSpeed)\$

motorPower: motorTorque*motorSpeed\$

fanPower: motorPower*powerEfficiency\$

ev(fanPower,
motorVolts=7.8,
fanSpeed=2500*%pi/30,
gearRatio=[4.65324181,4.678],
powerEfficiency=0.8,
specVolts=12,
freeSpeedSpec=18730*%pi/30,
stallTorqueSpec=100.54*0.00706155
)\$ float(%);

[20.00000014978008, 17.80773091738802]
``````

No takers?

Check your math.

Whoops, divided free speed by motor speed, that’s not a gear ratio… Okay, that should be a ~4.618 speed up. With the motor spinning at 541.4 RPM. And then melting down.

Also, my bonus problem is actually a lot easier, since there’s no quadratics involved and only one solution.

Using the Motor Calculator:

``````Motor Calculator  build MCALC16a 2/28/2016 1435

Select motor:
1)CIM     6)FP0673   d)FP2719  f)FP9012     g)FP9015    h)FP9013 j)MiniCIM
a)am-0912 b)am-0914  7)DensoL  8)DensoR     c)Denso0160 k)BAG    u)UserDefined
2)RS775Pro   3)RS540    4)RS550   e)RS775-12   5)RS775-18  m)RS555  n)am-0915
r)am-2193 s)am-2235  t)RS390   v)RS545      p)am-2161&2194
w)VEX2177hi  z)VEX2177std  q)UserFile  **2**

775Pro 217-4347 @ 12.00 volts:

@ free (no load):
oz-in      Nm     rpm    rpm%    amps   watts out  watts heat    eff%
0.0   0.000   18730   100.0     0.7         0.0         8.4     0.0

@ stall:
oz-in      Nm     rpm    rpm%    amps   watts out  watts heat    eff%
100.5   0.710       0     0.0   134.0         0.0      1608.0     0.0

@ max power:
oz-in      Nm     rpm    rpm%    amps   watts out  watts heat    eff%
50.3   0.355    9365    50.0    67.3       348.1       460.1    43.1

@ max efficiency:
oz-in      Nm     rpm    rpm%    amps   watts out  watts heat    eff%
6.8   0.048   17468    93.3     9.7        87.5        28.7    75.3

Select input:
1)oz-in  7)watts_in    3)rpm    5)amps   b)rpm&amps  8)eff%
2)Nm     6)watts_out   4)rpm%   9)volts  a)rpm&ozin  m)main menu   x)exit **9**

enter motor voltage: **7.8**

775Pro 217-4347 @  7.80 volts:

@ free (no load):
oz-in      Nm     rpm    rpm%    amps   watts out  watts heat    eff%
0.0   0.000   12175   100.0     0.5         0.0         3.5     0.0

@ stall:
oz-in      Nm     rpm    rpm%    amps   watts out  watts heat    eff%
65.4   0.462       0     0.0    87.1         0.0       679.4     0.0

@ max power:
oz-in      Nm     rpm    rpm%    amps   watts out  watts heat    eff%
32.7   0.231    6087    50.0    43.8       147.1       194.4    43.1

@ max efficiency:
oz-in      Nm     rpm    rpm%    amps   watts out  watts heat    eff%
4.4   0.031   11354    93.3     6.3        37.0        12.1    75.3

Select input:
1)oz-in  7)watts_in    3)rpm    5)amps   b)rpm&amps  8)eff%
2)Nm     6)watts_out   4)rpm%   9)volts  a)rpm&ozin  m)main menu   x)exit **6**

enter output Watts: **25**

775Pro 217-4347 @  7.80 volts:
oz-in      Nm     rpm    rpm%    amps   watts out  watts heat    eff%
62.4   0.441     **541**     4.4    83.2        25.0       624.3     3.9

775Pro 217-4347 @  7.80 volts:
oz-in      Nm     rpm    rpm%    amps   watts out  watts heat    eff%
2.9   0.021   **11633**    95.6     4.3        25.0         8.6    74.4
``````

11633/2500 = 4.65 gear motor speed DOWN

2500/541 = 4.62 gear motor speed UP

I’m assuming that you’re going to give the motor 12V now…

As the power efficiency is constant, and the power required to run the fan is a monotonically increasing function of fan speed, the maximum speed you can spin the fan is such that the motor is running at maximum power.

Max power out of the motor at 12V is 348.1W*, which is 13.92 times as much as in the original problem (20W/80%=25W). As the fan power requirement is proportional to the cube of the speed**, the fan is running 2.406 times as fast as the original problem, or 6,014 RPM. At max power output, the motor is running at 6087 RPM*. So the required gearing/belting is a 1.012:1 reduction. In practice, 1:1.

• These numbers from the previous post.

** From Kevin, but I’ll buy it because energy is proportional to mass times speed squared and mass per unit time is proportional to speed, making power proportional to speed cubed.

It’s not just a suggestion, it’s the law. I had reason to discover this when a client asked about upsizing a fan motor on a system someone else designed.

Your math is right for the RPMs at 12V, but you calculated your ratio at the 7.8V free speed/peak power. I realized I didn’t specify 12V or 7.8V, so I ran numbers for both:

``````@12V:
Motor Power = 347W
Fan Power = 347W * 80% = 277.6W
Power Ratio = 277.6W / 20W = 13.88
Speed Ratio = 13.88^(1/3) (cube root) = 2.403
Max Power Speed = 2500 * 2.403 = 6007.5 RPM
Gear Ratio = (18700 RPM / 2) / 6007.5 RPM = 3.113

@7.8V
Motor Power = 347W * (7.8/12)^2 = 146.6W
Fan Power = 146.6W * 80% = 117.3W
Power Ratio = 117.3W / 20W = 5.864
Speed Ratio = 5.864^(1/3) = 1.803
Max Power Speed = 2500 * 1.803 = 4508 RPM
Gear Ratio = (18700 * 7.8 / 12 / 2) / 4508 = 1.348

``````