DC motors are widely used in industry today due to their
low cost, higher efficiency than AC machines, less complex
control structure and wide speed and torque range. General
structures, properties, analysis, dynamical equations and
control of DC motors are given in many textbooks [1–3] and
some articles. Permanent magnet DC motors do not need field
excitation regulation and consume less input power. Due to
these advantages, permanent magnet DC motors have a wide
range of applications where constant speed will be maintained
at varying loads or different speeds will be obtained at
constant load [4,5]. In permanent magnet machines, there is
only one armature circuit and the flux produced by the
magnets is constant. Compared with conventional DC electric
machines, permanent magnet machines stand out with their
higher efficiency and simpler structures [6-10]. Due to the
absence of excitation windings and therefore no excitation
current, permanent-magnet machines offer high efficiency,
simple and robust structure and high power-to-weight ratio in
operation. The attractiveness of permanent-magnet machines
is further enhanced by the availability of high-energy rare-
earth permanent magnet materials such as SmCo and NdFeB
[11]. Besides, since there is no excitation circuit, it is not
possible to control the speed of a permanent-magnet DC motor
by changing the excitation current. However, since it is only
armature circuit, speed and torque control is done by
controlling this circuit and this simplifies the controller design.
With all these features, permanent magnet DC motors stand
out with their simpler structures and easy controllability when
compared to other DC motors.
In system analysis, there are two basic methods for
obtaining circuit equations: Modified Nodal Approach (MNA)
and State Variables Method (SVM) [12]. These methods have
advantages and disadvantages over each other. The most
important advantage of the equations of state method is that
the number of unknown variables is minimal. However, it is
very difficult to obtain the equations of the circuits with this
method. In the classical node method, all kinds of circuit
elements (such as voltage source, controlled sources) could
not be included in the system equations. To overcome these
drawbacks, the Modified Nodal Approach has been proposed
[13]. While the large number of unknowns of the generalized
node method is the disadvantage of the method, the
convenience in obtaining the equations is the most important
advantage of the method. [14-15]. The structure of Modified
Nodal Approach in the s-domain and t-domain are given
below in equation (1) and equation (2), respectively.
Here G, C, B are coefficient matrices. The frequency-
independent values resulting from obtaining all conductivity
and node equations form the G matrix, and the capacitance and
inductance values related to the frequency variable form the C
matrix. The U matrix is the vector containing the initial
conditions of independent current and voltage sources and
non-zero capacitances and inductances. Its unknown vector
contains both current and voltage variables. Considering
Complete Electrical Equivalent Circuit Based Modeling and Analysis
of Permanent Magnet Direct Current (DC) Motors
1ABDULLAH ALTAY, 2ALI BEKIR YILDIZ
1TÜBİTAK MRC, Vice Presidency of Energy Technologies, Renewable Energy Research Group 06800
Ankara, TURKEY
2Kocaeli University, Engineering Faculty, Electrical Engineering Department
41380 Kocaeli, TURKEY
Abstract: The modeling and analysis of permanent-magnet DC motor, whose attractiveness has increased with the
availability of high-energy rare earth permanent magnet materials such as SmCo and NdFeB, is very important in recent
years. The analysis of the established model is very important for researchers and engineers before siteworks. The most
important features of the established model are that the equations are simple to obtain, the model can be set up using a
simple platform, and ease of model analysis. Generally, for DC motor models, only the armature circuit model is given and
only the equations of the mechanical system are given. In the armature circuit model, equations are obtained by switching
to the s-domain, and analyzes are made on highly advanced platforms by using these equations. In this study, electrical
equivalents of mechanical equations were obtained and both armature circuit and mechanical parameters of a DC motor
were expressed as electrical circuit elements. Although the motor model includes mechanical parameters and variables, the
whole model is expressed only with electrical elements and variables. Thus, a complete electrical equivalent circuit is
proposed for the dynamic model of the permanent-magnet DC motor, in which both the armature and the mechanical part
can be modeled as an electrical circuit. With the analyzes, the performance of the permanent magnet DC motor model was
examined and it was seen that the system dynamics responses were compatible.
Keywords: Permanent Magnet DC Motor, Complete Electrical Equivalent Circuit, Analysis, Modeling
Received: July 21, 2021. Revised: June 25, 2022. Accepted: July 15, 2022. Published: August 2, 2022.
1. Introduction
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the variable types, the unknown vector can be split into
and vectors as shown in equation (3).
Here, the vector represents the node voltage
variables, and the vector represents the current
variables. Considering the unknown vector decomposed by
eq.(3), the structure of generalized nodal equations in the time
domain can be rearranged with equation (4) in the following
form.
In the studies examined in the literature [4-11], while DC
motor models are given, only the electrical circuit model of
the armature circuit is given. On the other hand, only the
equations of the mechanical system are used by expressing
them. No equivalent circuit models of these equations have
been produced. However, similar approaches were used for
the full electrical equivalent circuit model of the free-excited
DC motor in the study [16] and for the full electrical
equivalent circuit model of the PMSM in the study [17]. In this
study, analyzes will be made using the full electrical
equivalent circuit model and the MNA method. The most
important contribution of the model is the mechanical
equations and obtaining the electrical exact equivalent circuit
model of the position equation.
In this study, first of all, the electrical full equivalent circuit
of a permanent-magnet DC motor will be established. Then,
the model of the permanent-magnet DC motor will be
constructed using the generalized node equations structure in
the time domain given by equation (4). Numerical analyzes
based on the Backward-Euler method of the engine with the
MNA model will be made.
In Chapter II, armature, torque and motion equations of
permanent-magnet DC motor will be subtracted and dynamic
equations will be given. In Chapter III, the electrical exact
equivalent circuit of the permanent-magnet DC motor will be
constructed. In Chapter IV, the MNA model of the permanent
magnet DC motor with its full electrical equivalent circuit will
be given, and numerical analyzes based on the Backward-
Euler method regarding the established model will be carried
out for a sample motor. Transient stability analysis time
constants related to the system dynamics of the created MNA
model will be found, the duration of the transient state will be
determined, and the convergence step length will be
determined for numerical analysis. The compatibility between
the system dynamics outputs of the model and the calculated
values will be examined. At the same time, a performance
review of the established model will be made.
Due to the absence of excitation windings and therefore the
excitation current, permanent-magnet machines, which is
preferred because of their high efficiency in operation, their
low volume and weight, their high moment densities and
torque/weight ratios, have the armature equivalent circuit
shown in Figure 1.
La
Ra
Ea
Vm
Figure 1 Armature Equivalent Circuit Diagram of Permanent-
Magnet DC Motor
Equation of armature circuit,
(5)
(6)
Equation (6) is substituted in equation(5).
(7)
Moment equation:
(8)
(9)
Motion equation:
(10)
Equation (9) and Equation (10) taken together.
(11)
Position-velocity equation,
(12)
: Motor terminal voltage (V)
: Armature winding inductance (H)
: Armature winding resistance ()
: Armature current (A)
: Motor EMF Voltage (V)
: EMF constant - Motor design constant (V.s/rad)
: Motor air gap torque (Nm)
2. Equations of Permanent
Magnet DC Motors
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: Load torque (Nm)
: Motor shaft torque – Acceleration torque (Nm)
: Angular speed of motor rotor (rad/s)
: Motor friction constant (Nm.s/rad)
: Rotor inertia (kg./)
: Rotor position (rad)
By using equation (7), (11) and (12), the dynamic equations of
permanent magnet DC motor can be expressed as given in
equation (13).
=
(13)
When we look at the permanent magnet DC motor
mechanical equations, it is seen that the motor motion
equation can be modeled as a series RL circuit with the
following inferences.
Equation (7), which represents the armature circuit,
Series RL elements correspond to an electrical circuit
containing the dependent source and supply voltage.
This circuit equivalent is a very common notation.
An electrical circuit belonging to the equation o
f
motion given in Equation (10) will be obtained. Fo
r
this purpose, when equation (10) is examined
carefully, it can be seen that this equation can b
e
handled like the series RL circuit given in equatio
n
(14) due to its structure.
(14
)
An electrical circuit belonging to the mechanical
equation given in equation (11) will be obtained. Fo
r
this purpose, when equation (11) is examined
carefully, it can be seen that this equation can b
e
handled like the equivalent circuit in Figure 2 due to
its structure. It can be seen here that (Motor air gap
torque) value can be modeled as a dependent voltag
e
source and (Acceleration torque) value can b
e
modeled as a constant voltage source.
VTL
VTe=Kb.Ia
R1
LJ
Iω
Figure 2 Electrical Equivalent Circuit Diagram of Moment and
Motion Equations of Permanent-Magnet DC Motor
In these equations, it can be clearly seen that the
following transformations are made between the
electrical circuit and the mechanical system.
o
o
o
o
o
Likewise, the position-velocity equation from the motor
mechanical equations can also be modeled as an L-circuit, as
can be seen in Figure 3.
An electrical circuit belonging to the position-
velocity equation given in equation (12) will be
obtained. For this purpose, when equation (12) is
examined carefully, it is understood that this equation
is similar to the terminal equation of an inductance as
in equation (15). However, the terminal voltage will
be equivalent to a dependent voltage source as in
equation (16). In these equations, if
is taken, the value of corresponds to the current
and the value of to the current of the L circuit ().
As a result, the position-velocity equation is modeled
with the equivalent circuit in Figure 3.
(15)
(16)
Vθ=kθ.Iω Lθ
Iθ
+
Figure 3 Electrical Equivalent Circuit Diagram of Position-
Speed Equation of Permanent-Magnet DC Motor
o
o =1H
o
After these inferences, the proposed full electrical
equivalent circuit of permanent-magnet DC motor is given in
Figure 4. The first block of the fully electrical equivalent
circuit in Figure 4 expresses the armature circuit, the second
block expresses the moment and motion equations, and the
3. Complete Electrical Equivalent
Circuit Model of Permanent Magnet
DC Motor
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third block expresses the position-velocity equation. In the
fully equivalent circuit obtained, the variables that the
dependent sources depend on are also shown on the figure.
IaLa
Ra
Vm
Ea=Kb.Iω
VTL
VTe=Kb.Ia
R1
LJ
IωVθ=kθ.Iω Lθ
Iθ
+
Figure 4 Complete Electrical Equivalent Circuit Diagram of Permanent-Magnet DC Motor
In this section, the equations for the GDD method of the
full electrical equivalent circuit given in Figure 4 will be
given. Nodes and currents for the MNA model will be used as
shown in Figure 5.
La
Ra
Vm
aIRa
c
Ea=Kb.Iω
VTL
VTe=Kb.Ia
d
ef
ILa
ITe
ITL
IR1
Iω
Ia
b
R1
IVm
LJVθ=kθ.Iω Lθ
gIθ
IVθ
+
Figure 5 Nodes and Node Currents for the MNA Model
Main Equations;
Additional equations;
The equations obtained above, according to the system in
equation (2), the matrix and are given in
equation (17), (18), (19), and (20), respectively.
4. Mna Model of Permanent
Magnet DC Motor
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(17)
(18)
In order to examine the performance of the given model, a
sample motor will be examined with the parameters given in
Table 1.
Table 1 Permanent-magnet DC motor parameters
Here, first of all, unit conversions given in Chapter III are
made.
For the examination of system dynamics, we can consider
the characteristic matrix given in equation (21). Thus, the
dynamic behavior of the system can be predetermined by
equation (22). Equation (23) also calculates the transient time
(5τ).
(21)
(22)
(23)
Since for the circuit, it
can be deduced that it will be stable in the transient region and
vibratory damped motion since the circuit has complex roots
in the form of and .
In Figure 6, simulation results of variables are
given. For this study, it was ensured that the load torque and
value, which is the equivalent in the electrical full
equivalent circuit model, were as given in Table 2 below
during the simulation period.
Table 2 Load torque change for simulation study
t
0 V
0 Nm
100 V
100 Nm
It has been confirmed by the simulation results that the
transient time calculation calculated in Equation (23) is also
correct. ()
Figure 6 Graphics for The MNA Model
In Figure 7, simulation results of variables are
given.
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Figure 7 Graphics for The MNA Model
Figure 8 shows the graph that emerged as a result of the
simulation study. For this study, the value of was ensured
to be 200 V throughout the entire simulation process. The
transient region of value is examined and as expected,
value, which symbolizes motor speed, takes negative
values until value, which is dependent on value, is
greater than value.
Figure 8 Transient Graphic for The MNA
Model
In the study carried out, a very simple electrical equivalent
circuit of the permanent-magnet DC motor was extracted and
this equivalent circuit was modeled with MNA. In this study,
all variables and components in the mechanical equations of
permanent magnet DC motors are expressed in terms of
electrical variables and components. Thus, the complete
electrical equivalent circuit for the entire electromechanical
system of the DC motor is obtained. In the numerical example,
it is shown that various analysis and dynamic properties of DC
motor can be obtained with this electrical equivalent circuit. At
the same time, a temporary situation analysis of the MNA
model was made to the board and it was seen that the dynamic
results here were exactly as expected. The most important
features of the developed model are that the equations are
simple to obtain, the model can be set up using a simple
platform, and model analysis is quite easy. It is possible to
analyze a motor with a mechanical structure with programs that
solve electrical circuits.
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5. Conclusion
References
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Contribution of Individual Authors to the
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The authors equally contributed in the present
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problem to the final findings and solution.
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Conflict of Interest
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that are relevant to the content of this article.
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