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Formulae
Handbook
Jan Braun
V CC n L M L
maxon academy

Start
drive selection
Overview, system analysis,
preselection of controller & sensor
Chap. A.1
Motion of the load
Chap. A.2
Mechanical
transformation?
Mechanical drive design
Chap. A.3
Use of
gearhead?
Yes
Yes
Gearhead selection
Chap. 3.4
Motor type selection
Chap. 6.5
Winding selection
Chap. 6.5
Verification of controller & sensor
Chap. A.3
Drive selected
No
No
Selection process
Step 1
System
Overview
Ambient
conditions
v(t), F(t)
Load
Step 2
Load
Step 3
Drive
Gear Motor
Step 4
Gearhead
Communicaton
Step 5 + 6
Motor type
Winding
Step 7
Sensor
Controller
When performing a system analysis
overview
See chapter
A.1: Overview, system analysis
See chapter A.2: Motion of the load
mechanical transformation.
Mechanical drives
See chapter A.3:
Mechanical drives.
gearhead selection
maxon gearhead
See
chapter 3.4: maxon gearhead
types of
motorsSee chapter
6.5: Motor selection
selection of the winding
See chapter
6.5: Motor selection
Power
Controller

Foreword
This Formulae Handbook lists the most important formulae in relation to all components of
Numerous illustrations and the clear descriptions of the symbols on the respective page help the
reader to understand the formulae.
Roughly speaking, it is a collection of the most important formulae from the maxon catalog, as
for engineers, professors, lecturers and students, as a perfect supplement to the above mentioned
book.
Thank you
have read the manuscript and have given valuable suggestions for improvements. I also received

Content 5
A. Drive selection 6
1. Mass, force, torque 9
2. Kinematics
3. Mechanical drives
4. Bearing
5. Electrical principles
6. maxon motors
7. maxon sensor
8. maxon controller
9. Thermal behavior
10. Tables
11. Symbol list for the Formulae Handbook 56

A. Drive selection
A.1 Overview, system analysis
Mechanical design
J S
p
Set value
Feedback
J 2
Actuator
Sensor
Verify the power components
d1 mR Determine the boundary conditions
Cost considerations
l
d 2
J 1
Output
Measurement
d

A.2 Motion of the load
load requirements
Operating points
Velocity v
Speed n
4
1 2 3 4 1
Velocity v
Speed n
3
Mechanical play of the drive
2
0.4
0.2
1
Time t
Force F, torque M
-1500 -1000 -500 0 500 1000 1500
-0.2
Test torque mNm
Key data
Twisting angle °
Force F
Torque M
F /M max max
∆t max
0
-0.4
∆t tot
F RMS /M RMS
Time t
operating
points
mechanical play of the drive
key data

controller and sensor
Motion controller
Sensor
–
–
–
–
–
–
–

1. Mass, force, torque
1.1 Forces in general
Typical component forces in a drive system
m
m
F G
F H
F a
F G
F N
F N
F R
[F] = kg · m/s = kgm/s = N
F = m · a = m ·
v
a t
F = m · g
F H = F
F N = F
F R N
F
p F P F p
Symbol Name SI
F
F a
F
F H
F N
F p
F R
F
Symbol Name SI
a
g
k
m
p N/m bar

Calculating the total load force consisting of component forces
F L
F 1 F 2 F x
F 1
F L
F L
F 1
F x
Symbol Name SI
F L
F / F / F x
F 2
F 2
F L = F + F + ...+ F
F L = F
2 2
F = F + F L 1 2

1.2 Torques in general
General
r
F
M
Typical component torques in drive systems
r
M
M = J ·
30
·
M R = µ · F KL · r KL
= k m
Calculating the load torque consisting of component torques
M 1 M 2 M x
M L
M L
M 1
M x
M 2
Symbol Name SI
F
F
L
R
x
k m
L
L
Symbol Name SI
r
r
Symbol Name maxon

1.3 Moments of inertia of various bodies with reference
to the principal axes through the center of gravity S
Body type Illustration Mass, moments of inertia
Circular cylinder, disc 2 1
r2 J = r x 2
Hollow cylinder
Circular cone
Truncated circular cone
Circular torus
Sphere
Symbol Name SI
x
x
y
y
R
12
1
J = J = (3r y z 2 + 2 )
2 2 (r i )
a
2 2 J (r + ri )
x a
J = J =
y z 2
1
2
1
2 2
4
+ ri +
a 3
m =
J = x
J = J = (4r
y z 2 + h2 3
)
1
10
3
80 3
r2 r2
2 2 (r + r2
3
r + r )
2 1 1
1
J
3
x 10
r2 5 5 r1
3 3 r r1 2
m = 22r2
J = J =
1 2 2
x y 8
(4 + 5r )
4 1
J = (4 z 2 + 3r2 )
m =
4
3
r
2
5
3
J = J = J = r x y z 2
Symbol Name SI
h
m
r
r a
r i
r
r

Body type Illustration Mass, moments of inertia
Hollow sphere
Cuboid
Thin rod yy
Square pyramid
Arbitrary rotation body
Steiner's theorem
tion x r ss
Symbol Name SI
s
s
x
x
y
y
a a m
b b m
(r
4
3
a
3 3 i )
J = J = J
2
x y z 5
ra m =
1 2 2 J = x 12
( + )
m
12
1
J x = J z 2
1
3
20
1
20 1
J ( x
3
4
2 + 2 )
J ( y 2 + 2 )
5 5 i
3 3 r i a
2 m = f (x)
x1
x2
1
4 J = x 2
f (x)
x1
x2
2 J = m r + Js
x s
Symbol Name SI
c c m
h
l
m
r a
r i
r s s m
x
x x

2. Kinematics
2.1 Linear equations of motion
Uniform movement
Velocity v
s
t
v
Time t
Constant acceleration from a standing start
Velocity v
s
t
v
Time t
Constant acceleration from initial speed
Velocity v
s
t
v end
v start
Time t
Symbol Name SI
a
g
h
Remarks:
–
v =
t 2 1
2
h = · g · t 2 1
2
v end = v start + a
2 1
2
Symbol Name SI
v end
v start

2.2 Rotary equations of motion
General
2π rad 360°
1m
1rad
1m
1 rad = 57.2958°
1m
Uniform movement
Angular velocity,
Speed n
t
1m
,n
Time t
1 rad 360°
2
rad
of rotation
[] = rad/s
n =
[n] = /min
Constant acceleration from a standing start
Angular velocity,
Speed n
t
, n
Time t
[] = /s = rad/s
Constant acceleration from initial speed
Angular velocity,
Speed n
t
end , n end
start , n start
Time t
Symbol Name SI
end
30
30
t
30
t
30
Remarks:
–
– =
t
t
t
30
t
t 2
t
30
t
t 1
2
t
1
2 30
2 1
2
t
1
2 30
t
endstartt end start
30
n = n t
end start
t start 2
30
n = n t
end start
1
t start 2
1
nt
30 start 2 30
2 1
2
1
nt
30 start 2 30
Symbol Name SI
start
Symbol Name maxon
n end
n start

General Symmetrical
Suitability
Diagram
Task:
in time tot
at a
v max
v max
∆t a
∆s
∆tb ∆ttot
∆t c
v =
max
2
ac a = max vmax a a c
=
v
+ 2
max
vmax a = max a v max
∆s
∆t a ∆t b ∆t a
∆t tot
v =
max ( ) a
a max = ( a ) · a
= a
a max =
at a
=
a · max a
amax v = a · max max a
totv max
tot
a max
v max
a max
a c b
Symbol Name SI
a max
v max
v max
v max
a
a
· v
2
max
vmax a a = max max
a
v ( ) · v a max
max
a · ( ) · max a a
v = a · max max a
Symbol Name SI
a
b
c
tot

3/3 Trapezoidal Triangle
v max
v max = 1.5 ·
a max = 4.5 ·
∆s
∆t tot
2
= 1.5 ·
v max
a = 2 · max v2
max
=
3
2
·
amax ·
amax v = max
2
· · a · a max max
2
· · v
3
max
a max 3 · v max
a max
v max
v max = 2 ·
a max = 4 ·
∆s
∆t tot
2
= 2 ·
v max
a = max v2
max
= 2 ·
a max
v = · a max max
1
2
a = 2 · max v · · v max
max
2
9
3 1
· a · t max
v = · a · t · a t max max tot max tot
2
tot · amax t2 tot · a · max 2 1
4
1
2
v
amax 2
max
2 ·
vmax = 3 · amax
v
amax 2
max
vmax 2 · amax Symbol Name SI
a max
v max
v max = · a max ·
Symbol Name SI
tot

General Symmetrical
Suitability
Diagram
Task:
in time tot
n max
n max
∆t a
30
n = ·
max
∆
∆tb ∆ttot
of rotation
∆t c
2
a + c = max
2
a + c a 30 a + c = ·
n
+
2
max
nmax = max 30
·
ta at a
of max
=
30
totnmax · n ·
= · max n max
30 a
time tot max
of n max
max
Symbol Name SI
max
a
b
c
tot
a c + b
2
n max
n max =
∆
∆t a ∆t b ∆t a
∆t tot
30
·
( )
a
max = ( a ) · a
30
= n + a
max
·
nmax = max 30 a ·
= + max · a
a
n =
30
· · max max a
·( )
a
30 · nmax = · max n max
30 a max ( a ) a
n max = 30
· max · a
Symbol Name SI
Symbol Name maxon
n max

3/3 Trapezoidal Triangle
n max
30
n = 1.5 · ·
max
∆
∆t tot
= 4.5 · max 2 t tot
30
= 1.5 ·
· nmax n
= 2 · · max
2 max
2
302 3
= ·
2
·
max max n =
1
·
30
· · · · max max max
2
3 2
·
30
· · n max
= 3 · ·
max 30
n max
2
· · t
9 max 2
tot· t2 tot
n =
1
· · · · max 3
max max
30
30
n = 2 · max
·
ttot
= 4 · max 2 t tot
Symbol Name SI
max
tot
max
n max
30
n = 2 · ·
max
= 4 · max 2 t tot
= 2 ·
= max n2
max
30
·
∆
∆t tot
ttot
n max
· 2
30 2
= 2 ·
max 30
n = max · max
1
2
= · · · n
30 max
nmax = 2 · · max 30
· · t max 2
1
4
tot
n =
1 30
max 2
·
· max · n
max 2 max
= 30
·
nmax = 2 · 30
·
max Symbol Name maxon
n max

Notes

3. Mechanical drives
3.1 Mechanical transformation
Output power/transformation, general
Spindle
Ball screw
Trapezoidal screw
Eccentric drive
Crankshaft
Gearhead
Spur gearhed
Planetary gearhead
Bevel gear
Worm gear
Wolfrom gear
Linear
Rotation
Rack and pinion
Conveyor belt
Crane
Rover
Belt
Toothed belt
Chain drive
Special design
Cyclo gear
Harmonic Drive ®
Symbol Name SI
F
F L
L
in
mech
mech,in
mech,L
v
Output power, linear motion
[]
mech
Transformation, general
P mech,L =
Pmech,in
M in in
v F = L L
Output power, rotary motion
[] = s
30
P
mech
Transformation, general
P mech,L =
Pmech,in
M in in
M =
L L
Designations in the formulae
– L
– in
Symbol Name SI
v L
L
in
Symbol Name maxon
n

3.2 Transformation of mechanical drives, linear
Spindle drive
J S
p
Belt drive/conveyor belt/crane
m B
J 2
d 2
d 1
J 1
d 1
J 1
60
n = in p
· vL p FL M = ·
in 2
in a
m L S
M in S
in L
60
n = in
4 2
p 2
2 p
vL
d1 FL 2 M = in
d1
30
in a
J2 M = J + J +
1
Symbol Name SI
F L
in
X X
in
d X X m
d
d
m
2 d1 2 d2 +
JX
2
in L d1
2 d1 2 +
dX in a 2
m + m d L B 1
4
30
a Symbol Name SI
m L
m
p
v L
L
a
in
Symbol Name maxon
n in
in

Rack-and-pinion drive
Rover
p
m z
m F
J P , z
d
J W
60
n = v in p z L
M = in 2
p z FL
in a
p
M = J + J +
in, in P 2 z2 m + m L Z
in
4 a 2 30
M in =
in = L
60 vL n =
in d
2
p z
F L
d
2
in a m + m d L F
M = 4
a 2
J + J + W
30
Symbol Name SI
F L
in
in
d
m F
m L
m Z
p
= s
2
in L d
Symbol Name SI
v L
L
a
in
Symbol Name maxon
n in
in

Eccentric drive
e
J E
n in
v L (t) = · n in · e · sin · n in · t
30 30
m L
30 n F
a a L in
2
e cos
in L a
in L a
e
M = F F
L1 + F + + F
in,RMS a1 L2 a2
2
2 2 2 2
in a
M E
30
Symbol Name SI
F L
F L
F a
F a
F a
F a
in
in
a Symbol Name SI
in
e
m L
v L
t
a
Symbol Name maxon
n in
in

3.3 Transformation of mechanical drives, rotation
Gearhead
J 2
Belt drive
J 2
m R
d 2
i G J1
d 1
J 1
n in = n L · i
=
i · G
M in
M L
in a
M = J + J + 1 JL + J 2
2 2
i G ·
in L
i G
· ·
·
30
=
·
J + J + G 30 a
a
JL i · G
z + z 1 3
=
z1 d2 n = n ·
in L d1 M in =
d1 d2 M L
·
in a
M =
J in + J 1 + J L + J 2
in L
Symbol Name SI
in
L
x X
in
L
d x X m
d
d
J x
2
2
2
d d m d R · 1
1
1
· 2 + · 2 +
4 ·
·
30
·
d 2
= ·
d
d
2
1
d x
nin ta Symbol Name SI
i
m R
a
in
L
Symbol Name maxon
n in
n L
in

3.4 maxon gear
GP
Gearhead design type
GS Spur gearhead
GP Planetary gearhead
KD Koaxdrive
22
A
Diameter
Version
in mm
A Metal version
B Version with enlarged internal gear
C Ceramic version
HD Heavy Duty – for applications in oil
HP High power version
K Plastic version
L Low-cost version
M Sterilizable version for medical application
S Spindle drive with axial bearing
V Reinforced version
Operating ranges of gearheads
Z Low backlash version
Load speed nL Max. load speed
(determined by gearhead input speed)
n max,L
Continuous operation
M G,cont
Short-term operation
Symbol Name SI
i
M G,max
n max,L =
Load torque M L
n max,L
n max,in
i G
n max,in
Symbol Name maxon
n max,in
n max,L

4. Bearing
4.1 Comparison of characteristics of sintered sleeve bearings and ball bearings
Operating
modes
Speed
range
Radial /
axial load
Additional
operating
criteria
Bearing
play
of friction
Lubrication
Sintered sleeve bearings Ball bearings
–
–
–
–
–
–
–
–
–
–
–
r
–
–
–
–
–
–
–
–
–
–
–
– –
– –
–
–
Costs

Notes

5. Electrical principles
5.1 Principles of DC (Direct Current)
Electric power
[]
Power adjustment
RL = Ri P/P max
1.0
0.8
0.6
0.4
0.2
0.0
0.01
Ohm's law
0.1 1 10 100
R /R L i
+
_
I R i
P = U · I = R · I 2 U
=
R
2
V = R I
Linear source Consumer load
a
2 U0 P = max 4 · Ri
V
I
U
A
R
U
I =
R
U
R =
I
Symbol Name SI
I
max
V
R
U 0 U kl R L
I
4
2 · Ri =
b
Symbol Name SI
R i
R L
0
kl
I

Electrical time constant
U out
U in
100
%
60
40
20
63%
0
0 el 2 3 4 el el el
Pull-up / pull-down
R u
R d
+V
Pull-up
Logic
Logic
Pull-down
Open-collector output
Logic
Base
t
+V
R u
I out
Collector
Emitter
5 el
Open
Collector
Output
U out
Symbol Name SI
I out
L
R
R d
R u
=
L
el
R
[ el]
[ el]
el = R · C
Pull-up:
–
–
Pull-down:
–
–
Open-collector output (OC):
–
–
outI out · R u
Hall sensors
Symbol Name SI
t
in
out
+V
el

5.2 Electrical resistive circuits
Series resistor circuits
+
–
U
I
Parallel resistor circuits
+
–
U
I
R 1
I 1
Symbol Name SI
I
I , I
R
R 1
R 2
R 2
I 2
U 1
U 2
+...
R = R + R + ...
U1 =
U2
R 1
R 2
I = I + I +...
1 1 1
= + + ...
R R1 R2
R 1 · R 2
R =
R + R 1 2
I1 I2 =
R2 R1 Symbol Name SI
R , R

Voltage divider, no-load
+
–
U
I
R 1
R 2
Voltage divider, under load
+
–
I
+
U
–
R 1
U
R 2
I
R 1
R 2
I R2
Potentiometer
+
U
1
x
0
A
R 0
I R2
U L
A: Start
S: Wiper
E: End
U L
S
U 1
U L
I L
R L
R L
U 2
I L
R L
–
E
Winding resistance
RT = Rmot ·
Symbol Name SI
I
I L
I R
R
R
R , R
R L
R mot
R R R L
+
–
U 2 = U
U1 =
U2 R 1 + R 2
+
–
R 1
R x
R 1
R x
R1 R2 U2 I =
R2 R 2
U
=
R + R 1 2
R x
U L = U Rx + R 1
R L · R 2
R x = RL + R 2
R 2
I L = I · RL + R 2
R L
I R2 = I · RL + R 2
x · R 0 · R L
R = (x · R0 ) + R L
R
U L = U R + (1 – x) · R0
x
U = U L R0 2 (x – x ) +1
RL Symbol Name SI
R T T
L
x
Symbol Name Value

5.3 Principles of AC (Alternating Current)
Alternating quantities
T
Ohm's law
~
~
Resistances
Reactance
~
~
~
~
I
V U
Z
XL XL XC XC Impedance (AC resistance)
t
A
[f] =
[] = / s =
rad / s
f = T 1
tt
U(t)
I(t) =
Z
U(t)
Z =
I(t)
X L
X =
1
C
RL, or R R2 + X 2
Z =
Symbol Name SI
I
L
R
T
1
2
=
Symbol Name SI
X X or X L
X
X L
Z
f
t

General
f
1
f = C 2 · R · C
or fC = R
2 · L
Uout
Uin
f
R
U in C U out
L
U in R U out
I
I
U in
U in
U R
U out
U out
U L
U out
U in
1.0
0.5
0.2
0.1
90°
0.707
f = f C
45°
0°
0.1 0.2 0.5 1 2 5
f/f C
10
Uout Uin
f
C
U in R U out
R
U in L U out
I
I
U in
U in
U out
Symbol Name SI
I
L
R
in
L
U R
U C
U out
U out
U in
1.0
0.5
0.2
0.1
90°
0.707
f = f C
45°
0°
0.1 0.2 0.5 1 2 5
f/f C
10
=
1
1 + ( f / f C ) 2
Xc U = U out in 2 2 R + Xc
R
U = U out in 2 2 R + XL
Uout Uin 1 + ( f C / f ) 2
Symbol Name SI
out
R
X
X L
f
f
=
1
R
U = U out in 2 2 R + XC
X L
U = U out in 2 2 R + XL

6. maxon motors
6.1 General
What is special about maxon motors?
heartself-supporting ironless copper winding
–
–
–
–
–
–
–
–
maxon DC motor (brushed permanent-magnet energized DC motors)
RE program
–
–
–
–
A-max program
–
–
–
RE-max program
–
–
–
Properties of the two brush systems
Graphite brushes
–
–
–
–
–
–
–
–
Precious metal brushes
–
–
–
–
–
–
–
–

maxon EC motor
–
–
–
maxon EC range
–
–
–
–
EC-max range
–
–
–
–
EC-4pole range
–
– ®
–
–
–
–
–
–
–
maxon EC-i range
–
–
–

Electronic commutation
Commutation type Rotor position determination
Block commutation
Block-shaped phase currents
0° 60° 120°180° 240°300° 360°
Turning angle
Signal sequence diagram
for the Hall sensors (HS)
HS 1
HS 2
HS 3
Supplied motor voltage
(phase to phase)
+
U1-2
+
U2-3
+
U
3-1
0°
1
0
1
0
1
0
60 120 180 240 300 360
Sinusoidal commutation
Sinusoidal phase currents
0° 60° 120° 180° 240° 300° 360°
Turning angle
Comparison of DC and EC motors
DC motor (brushed)
–
–
–
–
0° 60° 120° 180° 240° 300° 360°
EMF
EC motor with HS Encoder
EMF
Legend
Star point
Time delay 30°
Zero crossing of EMF
EC motor (brushless)
–
–
–
–
–

6.2 Power consideration of the DC motor: in general
Motor as energy converter
P el = U mot mot
Power P
Speed n
n 0
P L
P L
M H
U mot > U N
U mot = U N
Torque M
Symbol Name SI
I mot
H
el
L
R mot
2
P = R J mot mot
elL 30
2
U · I = · n · M + R · I mot mot mot mot
P = n · M
L 30
Symbol Name SI
mot
N
Symbol Name maxon
n
n

6.3 Motor constants and diagrams
Motor constants
speed constant kntorque constant kM
Speed constant kn knn n = knind ind
Torque constant k M
k
I.
Information:
Dependence between k n and k M
Speed-torque line
mot
Speed n
n 0
M R
I 0
∆n
U mot > U N
∆M
U mot = U N
M H
I A
Torque M
Motor current I mot
Symbol Name SI
I mot
I
I
k
H
R
R mot
ind
mot
30 000
k · k = n M
rpm mNm
[ ·
V A ]
= 1
n n mot
H = k
R = k
rad
s · V
Nm
·
A
mot
n 30 000
=
M
·
R n mot 0
2 k MH
M
Symbol Name SI
mot
N
Symbol Name maxon
k n /
n
n

Voltage equation, motor
+
U mot
–
I mot
EMF
Efficiency
Speed n
n 0
max
L mot
R mot
U ind
U mot = U N
M H
Torque M
Symbol Name SI
I
I
I mot
k
L mot
H
R
R mot
N
ind
U = L · + R · I + U R · I + U
mot
0
·
30 000 Umotmot max =
( ) (withR M 0 )
1 –
I0 IA 2
Symbol Name SI
mot
max N
Symbol Name maxon
k n
n
n

6.4 Acceleration
Angular acceleration: Start with constant current
n 0
n0 nL Speed n
M , n L L
Speed n
l mot = constant
M H
t Time t
Torque M
M k I M mot
J J R L R L
J R L
30 M
Angular acceleration: Start with constant terminal voltage
n 0
n0 nL Speed n
M , n L L
Speed n
m
U mot = constant
t
M H
Time t
Torque M
Symbol Name SI
I mot
L
R
k
H
L
R
R mot
t
mot
max = JR + J L
J R L
30 kMI
M H
' · ln m
1 –
1 –
M + M L R · n0 MH M + M L R · n – n 0 L
MH
(J + J ) · R
' R L mot
= m
2 kM Symbol Name SI
max
m
m
L
Symbol Name maxon
n
n
n L

6.5 Motor selection
Motor type selection
Speed n
Continuous
operation
Max. permissible speed
M M RMS N
Short-term operation
M max
M H
Torque M
N Hmax
1 2 2 2 M = (t1 · M + t2 · M + ... + tn · M )
RMS ttot
1
2
n
Remark:
Winding selection
Speed n
kn
n0,theor Speed-torque line
high enough for all
n n + max
0,theor max
k = =
operating points
n n,theor Umot
n max
Deceleration Acceleration
M max
Safety
factor
~ 20%
Torque M
U mot
n max max
Recommendation:k n
k n
M max
I mot = I 0 + kM
Symbol Name SI
I mot
I
k
...n ...n
H
N
max
n
mot
Symbol Name SI
t …n ...n
t tot
Symbol Name maxon
k n
k n,theor
n
n max
n ,theor

7. maxon sensor
maxon incremental encoder
Channel A
Channel B
Signal edges (quadcounts)
Index channel I
360°e 90°e
Recommended applications QUAD MEnc MR EASY MILE Optical
✘ ✘ ✔ ✔ ✔ ✔
✔ ✔ ✔ ✔ ✔ ✘
✘ ✘ ✔ ✔ ✔ ✔
✘ ✘ ✔ ✔ ✔ ✔
✘, ✔ ✔ ✔ ✔ ✔ ✔
✘ ✘ ✘, ✔ ✔ ✔ ✔
✘ ✘ ✔ ✔ ✔ ✔
✔ ✔ ✔ ✔ ✔ ✘
✔ ✔ ✘ ✘ ✘ ✘
✘ ✘ ✘ ✔ ✔ ✔
✔ ✘ ✘ ✔ ✔ ✘
✔ ✔ ✔ ✘

Counts per turn from position resolution
N of the
360°
Remark:
Measurement resolution, motor speed
Example:
N
=
Q · N
= =
Q · N
qc
1
ms
qc
CPT
CPT
=
qc
= rpm
qc
Comment:
Symbol Name SI
N
i
Symbol Name SI
Symbol Name maxon

8. maxon controller
8.1 Operating quadrants
Operating quadrants
Quadrant II
Braking operation
Clockwise (cw)
Quadrant III
Motor operation
Counterclockwise
(ccw)
M
n
M
n
8.2 Selection of power supply
n
M
n
M
n
Quadrant I
Motor operation
Clockwise (cw)
Required supply voltage at given load (n L, M L)
V CC
M
Quadrant IV
Braking operation
Counterclockwise
(ccw)
0, 1-Q operation
–
–
–
–
4-Q operation
–
–
+ (maxon units)
L L max
Notes:
–
–
–
Achievable speed at given voltage supply
n (V U ) · (maxon units)
L CC max
n0,UN – L Symbol Name SI
L
N
V
max
U N
Symbol Name maxon
n
n L
n N

8.3 Size of the motor choke with PWM controllers
Calculation of current ripple
PWM scheme 1-Q 2-level (4-Q) 3-level (4-Q)
L tot
V CC
PP,max = 4 · Ltot · f PWM
L tot = L int + mot + L ext
PP,max = 2 · Ltot · f PWM
V CC
V CC
PP,max = 4 · Ltot · f PWM
L mot.
L mot .
– I N
I N
– I N
Calculation, additional external motor choke
PWM scheme 1-Q and 3-level (4-Q) 2-level (4-Q)
V CC
L ext = 6 · IN · f PWM
– L int – 0.3 · L mot
L ext
L ext >
DC amplifier
PWM
L int
L ext
DC
motor
L mot
Symbol Name SI
f
I N
L ext
L int
EC amplifier
PWM
PWM
PWM
V CC
L ext = 3 · IN · f PWM
1 /2 L int
1 /2 L int
1 /2 L int
1 /2 L ext
1 /2 L ext
1 /2 L ext
– L int – 0.3 · L mot
EC
motor
L mot
Symbol Name SI
L mot
L tot
V

9. Thermal behavior
9.1 Basics
Heat sources
Iron losses in EC
motors and motors
with iron core winding
P = V, magn 30
n · Mmagn
V,eddy
Resistance R
R
TW = R · Imot Joule power losses
Rmot in winding
R
Temperature T = Rmot · · T –
25°C TW Friction losses: in the bearings and in the brushes (brushed DC motors)
Losses in the
gearhead
Efficiency
100%
80%
60%
40%
M G, cont
1-stage
3-stage
5-stage
Torque M
Stall torque reduced through temperature rise
k 30
P = · n · M · (1 )
V,R mot mot G
P = · n · M ·
V,R 30 L L
1G G M HT = k M · I AT = k M · RTW
Storing heat
th v = c = cFemot = cFe Symbol Name SI
th
c
I T
I mot
k
t
HT T
L
magn
mot
m
m
m mot
m
V
V,eddy
U mot
Symbol Name SI
V,magn
V,R
R mot
R T
T
T
t
mot
Symbol Name maxon
n
n L
n mot
Symbol Name Value
c
c Fe

Analogy to electrical circuit:
Power loss P V Thermal Electrical Current I
C th,W
R th1
C th,s
R th2
T W
T S
∆T S
T A
∆T W
Thermal
Symbol Name Unit
V
T
Rth
Rth
U 1
U 2
Electrical
GND
Symbol Name Unit
I
V
V
V
R
R
F
F
Heating of a simple body Cooling of a simple body
T end
T start
Temperature T
th
63%
Heating
T max = 100%
–
t
th [ ]
= end · 1
Time t
Symbol Name SI
T
T
T end
T start
T
T
T start
T end
Temperature T
th
63%
t
th = start –
R 1
R 2
T max = 100%
Cooling
C 1
C 2
Time t
Symbol Name SI
t
max
t
th

9.2 Continuous operation
Motor: Winding and stator temperature
140
120
100
80
60
40
Temperature T
Heating of the winding in accordance
with the thermal time constant
of the winding W
Heating of the stator in accordance
with the thermal time constant
of the stator s
Thermal equilibrium
after several stator time
constants
Stator
20
Winding
Temperature
0
difference
1 10 100
Time t
1000 10000
Gearhead: Housing temperature
P V, G
P V, G
R th,G
T G
T A
∆T G
T W,∞
T S,∞
Symbol Name SI
I mot
R T
R th,
R th
R th
T
T
T
T
T A
R th1
R th2
∆T W
∆T S
= T = R th + R th
2
R · R · I th1 TA mot
= W 2
1Cu · R · R · I th1 TA mot
R th2
S = S – A = Rth1 + R th2
= T = R
W
R : R of
Symbol Name SI
T
T
T
t
Symbol Name Value

Permissible nominal current I N
n N
Speed n
Max. permissible speed
Continuous
operation
Determining R th2,mod
P V
T W
PV TS PV TA R th1
R th2, mod
∆T S
M N
I N
Short-term
operation
Torque M
Motor current I mot
I N,TA = I N ·
Symbol Name SI
I mot
I N
I T
N
V
R T
R th
R th
R th,mod
T
T – T max A
T – 25°C
max
I N,TA = I N ·
T – T R + R max A
th1 th2
·
T – 25°C R + R max th1 th2,mod
Motor under original conditions
–
Separate measurement during
continuous operation
I mot
– T
– T
2
1 · R · R · I Cu th1 TA mot
R · th2,mod S 2
RTA · I · (1Cu · )
mot
S
Symbol Name SI
T max
T
T
Symbol Name maxon
n
n N
Symbol Name Value

9.3 Cyclic and intermittent operation (continuously repeated)
90.00
Average
80.00
70.00
60.00
Time constant
winding 9s
end temparture,
winding
50.00
40.00
On Off On
(1s) (3s)
Off
30.00
0.1 1 10 100 1000 10000
Time t
Average temperature rise during intermittent operation
2
(R + R R I th1 th2 TA RMS
= W 2
1Cu (R + R R I th1 th2 TA RMS
Temperature T
Intermittent operation
Current I
Ion IRMS Temperature T
T W,av
T S,∞
t on
t off
Time t
T max
Symbol Name SI
I
I N
I T
I on
I
R T
R th
R th
T
T
R th2
S = (Rth1 + R th2 ) W
I RMS = I on ·
t on
t on + t off
I
T – T max A
I ·
on N T – 25°C max · ton – toff t on
I ont on
t off
2 I N
2 I on
T max A
T – 25°C
max
– 1 t on
Symbol Name SI
T max
T
T
t
t off
t on
Symbol Name Value

9.4 Short-term operation
140
120
100
Temperature T
80
60
40
20
≤ 0.1 s
Winding
Stator
Temperature
difference
0
1 10 100
Time t
1000 10000
T W,
T S,
T A
R th1
R th2
∆T W
Overload factor K
Meaning:
Imot K = ·
– T max IN – ton t on
t on
I mot
Symbol Name SI
I mot
I N
R T
R th
R th
T
T
T max
K =
T = T
= R th
2
R · R · I th1 TA mot
= W 2
1Cu · R · R · I th1 TA mot
T – 25°C R max th1
·
T – T R + R max S th1 th2
1
1 – exp – t on
W
K 2
t on W ln K 2 – 1
T max – T S
I mot = K · I N · Tmax – 25 C°
·
R th1 + R th2
R th1
Symbol Name SI
T
T
T
t
t on
–
Symbol Name Value

10. Tables
10.1 maxon Conversion Tables
General Information
meter m
ampere
Conversion example
Factors used for …
… conversions:
m
… gravitational acceleration:
… derived units:
Decimal multiples and fractions of units
of ten of ten
h
m
6 m
9 n
p
Power
mW
mW Torque
Moment of inertia ]
6
5
Mass Force
g p
g
6
Length
in ft m mm
m
mm
in
ft
Angular velocity ] Angular acceleration ]
rpm min rpm Linear velocity ]
m rpm
5
Temperature

Type of friction Friction condition Description

Examples

11. Symbol list for the Formulae Handbook
Name Symbol Unit Page number
a
F a
F a / F a
a
T
9
end start
in L
T
F
F p 9
N
m
I
i
I on
R I R
f
d m
X d X m
d m
d m
d m
l m 9
m
s r s m
F H 9
h m
...n t …n
e m
V,eddy W
el W
R / R / R
el
T end
T / T
R
R R L R x
F
f
F R 9
V,R W
R
T
g
th
h m
Z
ind
L
L ext
L int
r i m
R i
n in rpm
in
in
W
l m
abc abc m
I L
F L
F L / F L
R L
n L rpm
L
56

Name Symbol Unit Page number
v L
L
m
m L
m
m R
m Z
m
m mot
m F
m
m
a max
max
N max
n max,in rpm
n max,L rpm
T max
max W
n max rpm
max
max
v max
max
r m
rpm
mech,in W
L mech,L W
in
L L
mech W
m
L m‘
s s
x x
y y
X X
in
L
R
I mot
n mot rpm
mot
mot
I
n rpm
N n rpm
T I
I N
n N rpm
N
N
F N 9
t off
t on
r a m
I out
out
I , I
F / F / F x
R , R
x
T
F a
p m
x x / x m
x
W
V W

Name Symbol Unit Page number
V,magn W
p 9
R d
R u
f
r / r / r m
R m
X
X L
i
i
n ,theor rpm
k n,theor
T R T
R
I
v L
c
c
c Fe
n end rpm
n start rpm
rpm
in rpm
k n
rpm
p m
k 9
F 9
H
T HT
X or X L X
I
T I
T
J
+V
V
T
T start
t
L mot
R mot
kl
R
R th
R th,mod
R th
th
a / b / c a b c
...n ...n
k
magn
in in
k m
I
L tot
tot
t tot
v end
v start
L R
F 9
T R
T R
T

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