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Analysis of an electromechanical system
Final Exam
Ivan Mauricio Melo Melo
Estudiante de Ingeniería Mecatrónica
Universidad Autónoma de Occidente
Cali, Colombia
[email protected] - 2196155
In summary, this document includes the
mathematical model of an
electromechanical system that consists of
the displacement of an antenna, with a
position input, with the mathematical
model the transfer function is found, which
allows graphing the system to observe the
responses to a step entrance.
The position control system transforms the
position input signal into an output
response, also position, the position
control is found in many systems, such as
antennas, robotic arms, computer disk
drives, Among many more, in figure 1, an
example of them can be observed, which is
a system in which it has an azimuth output
that follows an input angle given by a
potentiometer, the latter converts said
angular displacement into a voltage, in the
same way the other potentiometer works
the same, and has a dc motor, a differential
amplifier and a power amplifier, and of
course, a load on the rotor.
Figure 1. Antenna system - layout.
Figure 2. Antenna system - schematic.
First, the mathematical model of the
engine is proposed:
𝑎
𝑎
𝑎
𝑎
[1]
𝑏
𝑚
[2]
𝑚
𝑡
𝑎
[3]
The previous equations are passed to
Laplace, giving as a result:
𝑎
𝑎
𝑎
𝑎
(𝑠) [1*]
𝑏
𝑚
(𝑠) [2*]
𝑎
𝑇
𝑚
(𝑠)
𝐾
𝑡
[3*]
We replace 3 * and 2 * in 1 *, giving an
equation such that:
( 𝑅
𝑎
+𝐿
𝑎
𝑆
) 𝑇
𝑚
( 𝑠
)
𝐾 𝑡
𝑏
𝑚
𝑎
(𝑠) [4]
Figure 3. Torque Tm.
With figure 3, the following equation is
obtained, which corresponds to the torque
𝑚
𝑚
𝑚
2
𝑚
𝑚
[5]
With equations 4 and 5, and considering
that L = 0, the transfer function of input
𝑎
and output 𝜃
𝑚
can be found:
𝑎
𝑡
𝑚
𝑚
𝑏
𝑚
𝑎
𝑎
𝑚
𝑎
𝑚
𝑎
𝑚
2
𝑚
𝑎
𝑏
𝑡
𝑡
𝑚
𝑎
2
𝑚
𝑚
𝑏
𝑡
𝑎
𝑡
𝑎
𝑚
𝑚
𝑎
𝜃
𝑚
( 𝑠
)
𝑒 𝑎
( 𝑠
)
𝐾
𝑡
𝑅 𝑎
𝐽 𝑚
𝑆[𝑆+
1
𝐽
𝑚
(𝐷
𝑚
𝐾
𝑏
𝐾
𝑡
𝑅
𝑎
)]
𝐾
𝑚
𝑆(𝑆+𝑎 𝑚
)
[6]
Equation 6 is the relationship between the
motor and the load.
Figure 4. Motor - torque relationship.
Considering that the equivalent inertia 𝐽
𝑚
and the equivalent viscous damping 𝐷
𝑚
as we see in figure 4, are given by:
𝑚
𝑎
𝑁
1
𝑁
2
2
𝐿
[7]
𝑚
𝑎
𝑁 1
𝑁
2
2
𝐿
[8]
Now, for the potentiometers, since the
input and output of the potentiometers are
configured in the same way, the transfer
functions of both will be the same, they are
represented by a relationship between the
voltage output and the angular positioning
input, the which is proportional:
𝑉 𝑖
(𝑠)
𝜃
𝑖
(𝑠)
𝑉
𝑁∗𝜋
𝑝𝑜𝑡
[9]
Now, we have the preamplifier, which
shows the transfer function when we have
the input 𝑉
𝑒
(𝑠), to obtain an output
𝑝
𝑉
𝑝
(𝑠)
𝑉 𝑒
( 𝑠
)
= 𝐾 [10]
In the introduction of the system, the
observation is made that:
𝑉
2 𝜋
[10*]
Figure 6. Response to a step.
In figure 6, the response of the transfer
function obtained in the development of
the system can be observed, with a given
step, which is reflected in the output by
means of a graph.
Figure 7. Map poles and zeros.
In figure 7 is the map of poles and zeros,
from which it can be induced, which will
have a slow response, since the pole is
very close to the origin, and this makes it
have a later response, as shown you can
see in the step response. On the other
hand, as it has no real part, no oscillations
will be observed.
Figure 8. Bode diagram.
In figure 8, you can see the graph which
shows us the magnitude in dB and the
phase in Deg, in relation to the frequency
given by Rad / s, also known as the bode
diagram.
Figure 9. Frequency for - 20dB
attenuation.
In figure 9, it was obtained from figure 8,
in which a frequency in which it has a
magnitude of - 20dB was searched, which
attenuates the system.
Figure 10. Transitory and stationary
regime.
In figure 10, we can see the representation
of the transitory and the stationary regime
which correspond, the transitory regime is
the space of time in which the function is
not in a constant response, but on the
contrary, it is varying, and consequently,
the stationary regime is when the signal is
at a constant which does not vary with
respect to the y-axis.
To conclude the analysis of this system,
the opportunity was given to use software
such as Matlab's, the feedback from some
other book recommended by the teacher,
in which they explain in a concrete way
the equations necessary to carry out the
exercise, on the other hand On the other
hand, it was possible to observe the
response of a position control system
which had not had the time to work with
one, the development of the system and
its respective analyzes became very
enjoyable.
References:
[1] Norman S, Nise - Control Systems
Engineering.
