H
Metrology
Measurement Science and Technology
ETH Zurich, Switzerland
Institute of Machine Tools and Manufacturing
H
H
H
Contact the Author
H
© Copyright
H
d0000722; rev02
Cooperation
H
H
Example
Direct-Current Motor with Permanent-Magnet
Karl H. Ruhm
Key Words
direct current, DC, motor, permanent-magnet, model, differential equation, derivation, torque, speed
Abstract
The sub-model of a DC motor concerning torque and speed is an ordinary linear differential equation of second order with one derivation in the input quantity.
Introduction
Depending on the demands, the model of a DC-motor with permanent-magnet generally is a multivariable
system (multiple input, multiple output; MIMO). Here we will consider only the dependence of speed (t) [s–1]
by the load torque ML(t) [N m]. So we treat only a monovariable system (single input, single output; SISO).
1
Differential Equation
RA d
1 cF 2  2
1 RA
1 d

(t)


(t)

(t)  
ML (t) 
ML (t) [s 3 ]
2
L A dt
 LA
 LA
 dt
dt
mit
2
d

s 1
rotational speed (  2n / 60)
ML
Nm
torque
RA

electrical resistance
LA
H  VA 1s inductivity

Nms2
massmoment of inertia
cF

machine value

Wb  V s
magnetic flux
MINIMOTOR SA
This is a linear ordinary differential equation (ODE) with two derivations of the output quantity (t) and one
derivation of the input quantity ML(t).
2
Signal Effect Diagram
This differential equation and its solution are shown in the signal effect diagram as follows:
M L (t)
d
dt
.
ω(0)
–
1
Θ
–
Σ
–
–
ω(0)
.
..
ω(t)
ω(t)
RA
LA
ω(t)
2
1
cF2Φ2
Θ LA
B1312
RA
Θ LA
Reliance Electric
If one is interested, not to have the derivation operation of an input quantity in the diagram, which may cause
problems with the implementation in a simulation program, a modification will lead to a differentiator-free realisation (Supplement → Module "Differentiator-Free Realisation").
Alternate Versions
There is a German version of this document: → d0000XXX
Modifications
Rev. Date
Modification
00
02.11.2006
First Edition
02
21.06.2012
Minor Additions
2