VM250 Design and Manufacturing I
Motor and Control
DC motors: introduction
A DC motor converts electrical energy to
mechanical energy, using magnetism to do this
Electrical power
I, V
Mechanical power
F, v or T, w
May contain permanent magnets (PM) or
electromagnets
Permanent Magnet DC motors are typically
used for low-mid power (< kW) applications
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Permanent magnet (PM) DC motors
A PM DC motor produces an output angular
speed proportional to applied voltage
May be used for velocity control without feedback (“open
loop”)
May be used for position, velocity, or torque control with
feedback (needs sensors)
Typically requires a motor driver, which
controls the current sent to the motor
Can have rotary and linear versions
Hart.
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Rotary
Linear
Components of a PM DC motor
Rotor (moving
component); typically
this is on a shaft.
Stator (stationary
component); typically
this consists of
permanent magnets
fixed to the motor
housing.
Housing
Bearings to support
the shaft
Brushes
Commutation
(switching)
components
Saitou, Hart, Image: Ilia Krivoruk (CC-BY-SA-3.0,
GFDL)
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Lorentz force
Force on current-carrying wire in magnetic field
F  i (I  B )
F: Lorentz force vector [N]; perpendicular to both B and I.
i: current [A]
I: vector of wire in magnetic field [m]
B: vector of magnetic flux density [Tesla]
Permanent
Magnet
N
5
-
S
i
i
B
Saitou
n
F
F
+
Armature (Coil)
Switching the current direction
Problem: cannot continue 360o revolution
unless we switch the current direction
Permanent
Magnet
F
N
F
n
N
S
i
i
B
F
n
i
S
i
B
F
-
+
Armature (Coil)
+
F
Cannot
rotate
further!!
F
F
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F
How does a DC motor work?
http://v.youku.com/v_show/id_XMjIzNzA3MzQw.html?from=s1.8-1-1.2
Image: Yves Pelletier
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Brushed DC motors
Solution to current switching: mechanical
“brushes”which are spring-loaded to stay in
contact with the commutator.
Commutator switches the
direction of the current, at just
the right point in the rotation of
the armature.
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Image: Ilia Krivoruk (Cc-by-sa-3.0,
GFDL)
Brushless DC motors
Instead of using a commutator, brushless motors use
electronics to switch the direction of the current.
The coils are rigidly attached to the stationary outer housing
The magnets are attached to the inner shaft, so they rotate
With a partner, discuss for two minutes: What would be
the advantages of a brushless motor? Any
disadvantages? (You don’t need to write; just discuss.)
Image: Sebastian Koppehel (CC BY 3.0)
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Torque-speed curve
 otors have (approximately) a linear torque-speed
relationship
n
n0
T  Ts  kn
Basic trend:
Slope = 1/k
Ts
T

Ts: stall torque (Torque at n=0) [Nm]

k: motor constant [N-m/rpm]
Tload ↑  n↓
Tload ↓  n↑
(some sources define the motor constant
differently)

n0: no-load speed (rotational speed at T =0) [rpm]

n: steady-state speed of revolution at the specified T (load) [rpm]
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Relationship between torque and
current
 Current (the red line) is proportional to torque (x-axis).
 If the motor is at rest, a small amount of current is required to
overcome friction before the motor will start moving.
Speed & Current vs. Torque
Image: Jong Min Park.
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Power
Output (mechanical) power = torque x speed.
For a DC motor with a linear torque-speed relationship,
maximum output power is obtained when the motor runs
at half the no-load speed. At this point, the torque is half
of the stall torque.
Power vs. Torque
Image: Jong Min Park.
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Efficiency
How do we define efficiency?
efficiency 
output mechanical power
T w
x100 
x100
input electrical power
VI
Efficiency vs. Torque
Image: Jong Min Park.
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DC motor example
Find steady-state speed (n) and current (iA) at
no-load condition
stall condition
K.Saitou; Image: Jong Min Park.
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DC motor example
No load condition
Image: Jong Min Park.
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DC motor example
No load condition
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Image: Jong Min Park.
DC motor example
Stall condition
Image: Jong Min Park.
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DC motor example
Stall condition
Image: Jong Min Park.
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In-Class Activity
Find steady-state speed (n) and current (iA) under these conditions:
a) Lifting a 10-oz load with a 2” radius pulley
b) Moving a 3-oz load with a 12” long arm. The arm weighs 2 oz.
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Image: Jong Min Park.
Solution to Part a)
Lifting a 10-oz load with a 2” radius pulley
T = r X F = 2 x 10 = 20 [in Oz]
Image: Jong Min Park.
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Solution to Part a)
Lifting a 10-oz load with a 2” radius pulley
T = r X F = 2 x 10 = 20 [in Oz]
Image: Jong Min Park.
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Solution to Part b)
Moving a 3-oz load with a 2-oz, 12” long arm
F= weight of apple; Fa = weight of arm
l = length of arm
T = l × F + l/2 × Fa = 12 x 3 + 6 x 2 = 48 [in. Oz]
Image: Jong Min Park.
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Solution to Part b)
Moving a 3-oz load with a 2-oz, 12” long arm
F= weight of apple; Fa = weight of arm
l = length of arm
T = l × F + l/2 × Fa = 12 x 3 + 6 x 2 = 48 [in. Oz]
Image: Jong Min Park.
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Dynamic considerations
So far, we thought about the operating torque
Torque required to operate the motor at a constant speed
This was determined by the steady load
Can obtain from static balance (as in the example problem)
What about the torque needed to reach the
operating torque?
This can be higher than operating torque, due to the extra
torque needed to accelerate the mass attached to the motor
(inertia)
Need to consider dynamics of the system
This will be important if you have a heavy machine and/or
have a lot of mass attached to your motor (large wheel, arm,
etc.)
See upcoming slides
Hart.
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Key points for motors
Understand how a DC motor works.
A DC Permanent Magnet motor has a linear
torque-speed curve.
Key values on this curve are the stall torque and
no-load speed.
Know how to do basic motor calculations (lifting
a load, etc). Also know how to use the torquespeed curve to find the value of speed for a
given load torque that you calculate.
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ME250 Design and Manufacturing I
Electric Motor Dynamic Analysis
Example
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Permanent magnet DC motors
Operation line for a fixed voltage
T  Ts  kn
Ts: stall torque (Torque at n=0) [Nm]
k: motor constant [Nm/rpm]
n0: no-load speed (rotational speed at T =0) [rpm] = Ts/k
n
n0
Slope = 1/k
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Ts
T
Permanent magnet DC motors
Effect of input voltage
V2
Ts 2  Ts1
V1
V2
n02  n01
V1
Eq. (1)
Assuming negligible internal friction
Too small voltage does not start the motor
Too large voltage burns out the motor
n
n02
n01
V2 > V1
V1
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Ts1
Ts2
T
Eq. (2)
Permanent magnet DC motors
Effect of gears (M:1 ratio with efficiency g)
Tsg  MTs
Tsg  g Tsg  g MTs
1
n0 g 
n0
M
Eq. (3)
Eq. (4)
g is typically 95% per single plastic gear; 5-30% for entire
gear box
n0
n
n0g
T
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Ts
T’sg
Tsg
Permanent magnet DC motors
Typical motor spec in catalog
h is maximum at 10-30% of Ts; 70%-90% of n0
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Motor driven vehicle: pusher
Scenario 1: Vehicle that can push with force F
Required driving torque (factor of safety fs is typically 2)
TD  f s Fr
Eq. (5)
Stall torque of motor-gear system should be larger than this
TD  g MTs
TD  Tsg
Eq. (6)
F
TD
r
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Assuming no slipping/tipping: check these first!
Motor selection: pusher
1. Calculate TD using Eq. (5).
2. Select a candidate motor and read Ts and n0 from
datasheet. Multiply Ts with the number of motors
if more than one is used. If all motors have been
selected, reduce fs and go to step 1.
3. Scale Ts and n0 to input voltage using Eqs. (1)
and (2).
4. Assuming motor operates at maximum efficiency
(=0.2Ts), calculate required torque Tr by:
Tr  0.2g Ts
5. Calculate required gear ratio Mr by:
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TD
Mr 
Tr
Motor selection: pusher
6. Select a gear ratio M which is close to Mr and is
available to the candidate motor. If all gear
ratios have been selected, go to step 2.
7. Check if Eq. (6) is satisfied. If not, go to step 6.
8. Get operation line equation of motor-gear
system using Eqs. (3) and (4).
9. Calculate n at T = TD on the operation line of
motor-gear system and check the value and the
closeness to the max efficiency (~80% of n0g).
If not satisfactory, go to step 6.
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Motor driven vehicle: runner
Scenario 2: Vehicle that can move distance d in
time tf from dead stop
Motor does not produce constant torque  cannot use
constant acceleration formula!
Must start with the equation of motion
Assuming no slipping/tipping: check these first!
n
The vehicle is accelerating to the right. The red arrow (ma) is
an inertial force that is acting on COG of the vehicle *in the
opposite direction of acceleration*.
n0
TD
ma
Slope = 1/k
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Ts
T
r
Motor driven vehicle: runner
Recall: Location, velocity, and acceleration
ds
v
dt
dv
a
dt
Equation of motion of vehicle
F
ma   FL
fs
FL = steady load of vehicle (more about this later)
factor of safety fs is typically 2
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Motor driven vehicle: runner
Operation line of motor-gear system with respect
to vehicle motion
F  Fsg  kvv
Where:
kv 
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Fsg 
Fsg
v0 g
Tsg
r

g MTs

r
2
r
n0 g
60
g MTs
Eq. (7)
r
30g M Ts

2
 r n0
2
Eq. (8)
Motor driven vehicle: runner
Eliminate a and F from equation of motion
Fsg  f s FL
kv
dv

v
dt
fsm
fsm
Solve with initial condition v=0 at t=0:
Fsg  f s FL
v
kv
v
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
 kv  
t 
1  exp  

 f s m  
steady state speed v= (Fss-fsFL)/k’v
t
Motor driven vehicle: runner
Travel distance in tf from dead stop
Fsg  f s FL
s   vdt 
kv
0
tf

 kv
 
f s m 
t f   1
t f 
exp  
kv 

 f s m  
This must be larger than required distance of
travel
Fsg  f s FL
kv

 kv
 
f s m 
t f   1  d
t f 
exp  
kv 

 f s m  
How to get FL??
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Eq. (9)
Motor driven vehicle: runner
Experiment needed for measuring FL…
Vehicle without motor-gear system (it is accounted in g)
Measure force during steady-state motion
May be negligibly small -- Depends on how well you build it.
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Motor selection: runner
1. Calculate average rotational speed by:
60 d
n
2 t f r
2. Select a candidate motor and read Ts and n0
from datasheet. Multiply Ts with the number of
motors if more than one is used. If all motors
have been selected, reduce fs and go to the start
of step 2.
3. Scale Ts and n0 to input voltage using Eqs. (1)
and (2).
4. Assuming motor operates at maximum
efficiency, calculate operating speed nr by:
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nr  0.8n0
Motor selection: runner
5. Calculate required gear ratio Mr by:
nr
Mr 
n
6. Select a gear ratio M which is close to Mr and is
available to the candidate motor. If all gear
ratios have been selected, go to step 2.
7. Check if Eq. (9) is satisfied. If not, go to step 6.
8. Calculate steady-state speed ng∞ by:
ng 
1

M

f s FL r 
1 
 n0
 g MTS 
9. Get operation line equation of motor-gear
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system using Eqs. (3) and (4)
Motor selection: runner
10.Calculate T at n = ng∞ on the operation line and
check the value and closeness to the max
efficiency (20% of T’sg). If not satisfactory, go
to step 5.
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Summary
Permanent magnet DC motor
Operation line, stall torque, no torque speed
Effect of input voltage and gears on operation line
Motor-driven vehicle
Pusher
Runner
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