### 1_2_6_A MaximumMotorPower

```Maximizing Motor Power at
Constant Voltage
Principles Of Engineering
When a motor is loaded with no torque,
it spins at no-load speed, w0 .
When a motor is loaded with the stall torque
tstall or more, the motor will stop.
tstall = stall torque
Speed w
Speed vs. Torque is Linear
w0
Half speed
Stall
−
=
∙  +

tstall
Torque t
Speed w
Optional slide for the genius:
Speed vs. Torque Depends on Voltage
Torque t
Power
Power = Torque x Angular Speed
P=t•w
Speed w
Power = Torque x Speed
w0
No power. Why?
No power. Why?
tstall
Torque t
Speed w
Maximum Power at ½ Stall Torque
w0
−
=
∙  +

Maximum power.
Why?
1
tstall
2
Torque t
tstall
Why half stall torque?
−
=
∙  +

Linear equation
from last slide.
Multiply
both
−
=
∙  +  () sides by torque,

product on left is
power.
−
=
∙  +

Down-facing
parabola.
Maximum Power at 1/2 Stall Torque
Power P
−
=
∙  +

Maximum power
1
tstall
2
Torque t
Stall
VEX Application: Pick a Gear Ratio
Step 1. This motor is stalled. What is the
stall torque?
tstall = d x F┴
= (3 in.)(1.4 lb)
= 4.2 lb·in.
VEX motor
Step 2. At what torque will the motor deliver
maximum power? tmaxPower = ½ tstall
= ½ (4.2 lb·in.) = 2.1 lb·in.
VEX Application: Pick a Gear Ratio
Step 3. The same motor as in
the last slide is being used to
power a winch with a drum of
radius 2 in. lifting 0.2 lb. What
torque is the motor applying?
tout = d x F┴
F = 0.2 lbs
= (2 in.)(0.2 lb)
= 0.4 lb·in.
VEX Application: Pick a Gear Ratio
drive, what gear ratio would
make the motor deliver
maximum power?
GR =
τout 0.4 in.∙ lb
=
τin
2.1 in.∙ lb
F = 0.2 lbs
12 teeth
= 0.19 ≐
60 teeth
VEX Application: Pick a Gear Ratio
Now the winch is geared so
that the motor is delivering
half its stall torque, for
maximum power.
GR
GR==
τout 0.4 in.∙ lb
=
τin
2.1 in.∙ lb
12 teeth
= 0.19 ≐
60 teeth
Human Application: Bicycle “gears”
Can you explain why changing “gears”
on a bike helps a person win a race
that goes up and down hills?
References
MIT Center for Innovation in Product Development (1999).