2. Physical System
A DC motor converts direct-current (DC) electrical energy into rotational mechanical energy. A
major fraction of the torque generated in the rotor (armature) of the motor is available to drive an
external load. DC motors are widely used in numerous control applications because of features
such as high torque, speed controllability over a wide range, portability, well-behaved speed-
torque characteristics, and adaptability to various types of control methods. DC motors are
classified as either integral-horsepower motors (≥ 1 hp) or fractional-horsepower motors (< 1
hp). Within the class of fractional-horsepower motors, a distinction can be made between those
that generate the magnetic field with field windings (an electromagnet) and those that use
permanent magnets. In industrial DC motors, the magnetic field is usually generated by field
windings, while DC motors used in instruments or consumer products normally have a
permanent magnet field.
The physical system, shown in Figure 2 and typical of commonly used motors; is a fractional-
horsepower, permanent-magnet, DC motor in which the commutation is performed with brushes.
The load on the motor is a solid aluminum disk with a radius r = 1.5 inches (0.0381 m), a height
h = 0.375 inches (0.0095 m), and a moment of inertia about its axis of rotation J
load
=
1
(where mass
and density ρ = 2800 kg/m
2
2
mr
m r h
=ρπ
2
3
) which equals 8 8 10
5
.
×
−
kg-m
2
. The system
is driven by a pulse-width-modulated (PWM) power amplifier. Motor speed is measured using
an analog tachometer.
Figure 2. Physical System
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DC Motor / Tachometer Closed-Loop Speed Control System 5
The DC motor/tachometer system has a single input and a single output. The input is the voltage
applied across the two motor terminals. The output is the voltage measured across the two
tachometer terminals.
The principle of operation of a DC motor is illustrated in Figure 3. Consider an electric
conductor placed in a steady magnetic field perpendicular to the direction of the magnetic field.
The magnetic field flux density
B is assumed constant. A DC current i is passed through the
conductor and a circular magnetic flux around the conductor due to the current is produced.
Consider a plane through the conductor parallel to the direction of flux of the magnet. On one
side of this plane, the current flux and field flux are additive; on the opposite side, they oppose
each other. The result is an imbalance magnetic force
F on the conductor perpendicular to this
plane. This force is given by
Fid
FBi
B
=
×
=
∫
G
G
G
A
A
v
where
B = flux density of the original field
i = current through the conductor
A = length of the conductor
Figure 3. Operating Principle of a DC Motor
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DC Motor / Tachometer Closed-Loop Speed Control System 6
The active components of B, i, and F are mutually perpendicular and form a right-handed triad.
If the conductor is free to move, the force will move it at some velocity
v in the direction of the
force. As a result of this motion in the magnetic field
B, a voltage e
b
is induced in the
conductor. This voltage is known as the back electromotive force or back e.m.f. and is given by
b
evBdB=×⋅=v
∫
G
JJK
G
AA
The flux due to the back e.m.f. will oppose the flux due to the original current through the
conductor (Lenz's Law), thereby trying to stop the motion. This is the cause of electrical
damping in motors. Saying this another way, the back e.m.f voltage tends to oppose the voltage
which produced the original current.
Figure 4. Elements of a Simple DC Motor
Figure 4 shows the elements of a simple DC motor. It consists of a loop, usually of many turns
of wire, called an armature which is immersed in the uniform field of a magnet. The armature is
connected to a commutator which is a divided slip ring. The purpose of that commutator is to
reverse the current at the appropriate phase of rotation so that the torque on the armature always
acts in the same direction. The current is supplied through a pair of springs or brushes which
rest against the commutator. Figure 5(a) shows a rectangular loop of wire of area A = d
l
carrying a current
i and Figure 5(b) shows a cross-section of the loop.
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DC Motor / Tachometer Closed-Loop Speed Control System 7
Figure 5. Rectangular Loop of Current-Carrying Wire in a Magnetic Field
From Figure 5(b), the torque of the motor is given by:
()
n
d
T 2F N iB sin dN iABNsin mBNsin
2
==θ=θ=
A θ
G
G
G
TNmB
=
×
where
N is the number of turns of the armature,
=
A
A
d
is the area of the armature, d/2 is the
moment arm of the force
F
n
on one side of a single turn of wire, and the magnetic moment
, which is a vector with a direction normal to the area A (i.e.,
n
direction with the
direction determined by the right-hand rule applied to the direction of the current).
miA=
ˆ
The motor used in this system is the Honeywell 22VM51-020-5 DC Motor with Tachometer.
The factory specifications are shown in Table 1.
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DC Motor / Tachometer Closed-Loop Speed Control System 8
Table 1. Specifications of the Honeywell 22VM51-020-5 DC Motor with Tachometer
Motor Characteristics Units Values Parameter
Rated Voltage (DC) volts 24 -
Rated Current (RMS) amps 2.2 -
Pulsed Current amps 13.4 max -
Rated Torque N-m 9.18E-2 -
Rated Speed RPM 2225 -
Back EMF Constant volts-sec/rad 0.0374 K
b
Torque Constant N-m/amp 0.0374 K
t
Terminal Resistance ohms 3.8 R
Rotor Inductance henry 6.0E-4 L
Viscous Damping Coefficient N-m-sec/rad 6.74E-6 B
Rotor Inertia (including Tach) kg-m
2
3.18E-6 J
m
Static Friction Torque N-m 9.18E-3 T
f
Tachometer Voltage Constant volts-sec/rad 0.0286 K
tach
The power amplifier used in this system is the Advanced Motion Controls Model 25A8. It is a
PWM servo-amplifier designed to drive brush-type DC motors at a high switching frequency.
The factory specifications for this amplifier are shown in Table 2.
Table 2. Specifications of the Advanced Motion Controls Model 25A8 PWM Amplifier
Power Amplifier Characteristics Values
DC Supply Voltage 20-80 V
Maximum Continuous Current
± 12.5 A
Minimum Load Inductance
200
µH
Switching Frequency
22 Khz
± 15%
Bandwidth 2.5 KHz
Input Reference Signal
± 15 V maximum
Tachometer Signal
± 60 V maximum
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DC Motor / Tachometer Closed-Loop Speed Control System 9
Two power supplies are used to drive the system. Their specifications are shown in Table 3.
Table 3. Specifications of Power Supplies
EMCO Supply Model PSR-4/24 Values
Supply voltage +24 Volts
Maximum Continuous Current 4 amps
Maximum Peak Current 5 amps
Proto-Board 203A Values
Supply Voltage
± 15 V @ 0.5 A
Supply Voltage 5 V @ 1.0 A
A schematic diagram of a DC motor is shown in Figure 6.
Figure 6. Schematic Diagram of a DC Motor
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DC Motor / Tachometer Closed-Loop Speed Control System 10
Motion transducers that employ the principle of electromagnetic induction are termed variable-
inductance transducers. When the flux linkage (defined as the magnetic flux density times the
number of turns in the conductor) through an electrical conductor changes, a voltage is induced
in the conductor. This, in turn, generates a magnetic field that opposes the primary field. Hence,
a mechanical force is necessary to sustain the change in flux linkage. If the change in flux
linkage is brought about by relative motion, the mechanical energy is directly converted
(induced) into electrical energy. This is the basis of electromagnetic induction, and it is the
principle of operation of electrical generators and variable-inductance transducers. Note that in
these devices, the change in flux linkage is caused by mechanical motion, and mechanical-to-
electrical energy transfer takes place under near-ideal conditions. The induced voltage or change
in inductance may be used as a measure of motion. Variable-inductance transducers are
generally electromechanical devices coupled by a magnetic field.
There are three primary types of variable-inductance transducers:
a)
Mutual-inductance transducers, e.g., linear variable differential transformer (LVDT), rotary
variable differential transformer (RVDT), mutual-induction proximity probe, resolver,
synchro-transformer.
b)
Self-induction transducers, e.g., self-induction proximity sensor.
c)
Permanent-magnet transducers, e.g., permanent-magnet DC velocity sensors (DC
tachometers), AC permanent-magnet tachometers, AC induction tachometers.
The permanent-magnet DC velocity sensor (DC tachometer) is a variable-inductance transducer.
It has a permanent magnet to generate a uniform and steady magnetic field. A relative motion
between the magnetic field and an electrical conductor induces a voltage that is proportional to
the speed at which the conductor crosses the magnetic field. In some designs, a unidirectional
magnetic field generated by a DC supply (i.e., an electromagnet) is used in place of a permanent
magnet.
The principle of electromagnetic induction between a permanent magnet and a conducting coil is
used in speed measurement by permanent-magnet transducers. Depending on the configuration,
either rectilinear speeds or angular speeds can be measured. Schematic diagrams of the two
configurations are shown in Figure 7. These are passive transducers because the energy for the
output signal
v
0
is derived from the motion (measured signal) itself. The entire device is usually
enclosed in a steel casing to isolate it from ambient magnetic fields.
In the rectilinear velocity transducer, the conductor coil is wrapped on a core and placed
centrally between two magnetic poles, which produce a cross-magnetic field. The core is
attached to the moving object whose velocity must be measured. The velocity
v is proportional
to the induced voltage
v
0
. A moving-magnet and fixed-coil arrangement can also be used, thus
eliminating the need for any sliding contacts (slip rings and brushes) for the output leads, thereby
reducing mechanical loading error, wearout, and related problems.
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DC Motor / Tachometer Closed-Loop Speed Control System 11
Figure 7. Permanent-Magnet Transducers:
(a) rectilinear velocity transducer; (b) DC tachometer-generator
The tachometer-generator is a very common permanent-magnet device. The rotor is directly
connected to the rotating object. The output signal that is induced in the rotating coil is picked
up as DC voltage
v
0
using a suitable commutator device - typically consisting of a pair of low-
resistance carbon brushes - that is stationary but makes contact with the rotating coil through
split slip rings so as to maintain the positive direction of induced voltage throughout each
revolution.
The induced voltage is given by
0 c tach c
vBvB(2hn)(r)K== ω=A ω
for a coil of height
h, radius r, and n turns, moving at an angular speed
ω
c
in a uniform magnetic
field of flux density
B. This proportionality between v
0
and
ω
c
is used to measure the angular
speed
ω
c
.
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DC Motor / Tachometer Closed-Loop Speed Control System 12
3. Physical Model
The challenges in physical modeling are formidable:
• Dynamic behavior of many physical processes is complex.
• Cause and effect relationships are not easily discernible.
• Many important variables are not readily identified.
• Interactions among the variables are hard to capture.
The first step in physical modeling is to specify the system to be studied, its boundaries, and its
inputs and outputs. One then imagines a simple physical model whose behavior will match
sufficiently closely the behavior of the actual system. A physical model is an imaginary physical
system which resembles the actual system in its salient features but which is simpler (more
"ideal") and is thereby more amenable to analytical studies. It is not oversimplified, not overly
complicated - it is a slice of reality. The astuteness with which approximations are made at the
outset of an investigation is the very crux of engineering analysis. The ability to make shrewd
and viable approximations which greatly simplify the system and still lead to a rapid, reasonably
accurate prediction of its behavior is the hallmark of every successful engineer. This ability
involves a special form of carefully developed intuition known as
engineering judgment. Table
4 lists some of the approximations used in the physical modeling of dynamic systems and the
mathematical simplifications that result. These assumptions lead to a physical model whose
mathematical model consists of linear, ordinary differential equations with constant coefficients.
Table 4. Approximations Used in Physical Modeling
Approximation Mathematical Simplification
Neglect small effects Reduces the number and complexity of the
equations of motion
Assume the environment is independent of
system motions
Reduces the number and complexity of the
equations of motion
Replace distributed characteristics with
appropriate lumped elements
Leads to ordinary (rather than partial)
differential equations
Assume linear relationships Makes equations linear; allows
superposition of solutions
Assume constant parameters Leads to constant coefficients in the
differential equations
Neglect uncertainty and noise Avoids statistical treatment
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DC Motor / Tachometer Closed-Loop Speed Control System 13
Let's briefly discuss these assumptions:
Neglect Small Effects
Small effects are neglected on a relative basis. In analyzing the motion of an airplane, we are
unlikely to consider the effects of solar pressure, the earth's magnetic field, or gravity gradient.
To ignore these effects in a space vehicle problem would lead to grossly incorrect results!
Independent Environment
Here we assume that the environment, of which the system under study is a part, is unaffected by
the behavior of the system, i.e., there are no loading effects. In analyzing the vibration of an
instrument panel in a vehicle, for example, we assume that the vehicle motion is independent of
the motion of the instrument panel. If loading effects are possible, then either steps must be
taken to eliminate them (e.g., use of buffer amplifiers), or they must be included in the analysis.
Lumped Characteristics
In a lumped-parameter model, system dependent variables are assumed uniform over finite
regions of space rather than over infinitesimal elements, as in a distributed-parameter model.
Time is the only independent variable and the mathematical model is an ordinary differential
equation. In a distributed-parameter model, time and spatial variables are independent variables
and the mathematical model is a partial differential equation. Note that elements in a lumped-
parameter model do not necessarily correspond to separate physical parts of the actual system. A
long electrical transmission line has resistance, inductance, and capacitance distributed
continuously along its length. These distributed properties are approximated by lumped
elements at discrete points along the line.
Linear Relationships
Nearly all physical elements or systems are inherently nonlinear if there are no restrictions at all
placed on the allowable values of the inputs. If the values of the inputs are confined to a
sufficiently small range, the original nonlinear model of the system may often be replaced by a
linear model whose response closely approximates that of the nonlinear model. When a linear
equation has been solved once, the solution is general, holding for all magnitudes of motion.
Linear systems also satisfy the properties of superposition and homogeneity. The superposition
property states that for a system initially at rest with zero energy, the response to several inputs
applied simultaneously is the sum of the individual responses to each input applied separately.
The homogeneity property states that multiplying the inputs to a system by any constant
multiplies the outputs by the same constant.
Constant Parameters
Time-varying systems are ones whose characteristics change with time. Physical problems are
simplified by the adoption of a model in which all the physical parameters are constant.
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DC Motor / Tachometer Closed-Loop Speed Control System 14
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