PI Current Controller Design for DC Drive System using 4- Quadrant
DC-DC Converter and 3- Phase Rectifier
MUSTAFA SAAD, KHALED MUSTAFA
Control Engineering Department,
College of Electronic Technology Bani Walid,
Bani Walid,
LIBYA
Abstract: - DC Motors is considered one of the important machines in control systems such as industrial robots,
vehicles, and process control. Current control of a DC drive is very desirable because by controlling the current,
the torque is controlled. Moreover, current control can be used to prevent large damaging armature currents
during start-up. A good current loop is very important when setting up a DC drive. When the control of the
current loop is good, the steady state motor current should respond exactly with the reference current, and the
transient response to the step change in the reference current should be fast and well-damped. This paper
presents two different types of current controller converters that are commonly used in DC motor drives. The
first converter is a 4-quadratic switch-mode DC-DC converter, while the second controller is a 3-phase
controller rectifier. Both current converters were simulated using Matlab Simulink, where a PI current
controller for the inner loop current control DC motor controls these converters. The PI controller was designed
based on the Bode plot frequency response method. In this research, the design procedure was based on the
small signal model and then, verified using a large signal model. The simulation result showed that the
performance using a 4-quadrant DC-DC converter gave better performance than a 3-phase rectifier.
Key-Words: - DC Motor Drive, 4-quadrant Switch Mode, 3-phase Rectifier, Closed Loop, PI Controller
Received: March 27, 2023. Revised: January 7, 2024. Accepted: February 22, 2024. Published: April 2, 2024.
1 Introduction
DC motors were used last time. The DC network
was the first developed electric network and was
constructed to work on the DC electric network.
Nowadays, the majority of industry-installed motors
consist of AC motors, due to their high-speed
operation and their smaller volume and weight.
Additionally, AC motors, due to their construction,
require lighter maintenance and are cheaper
compared to DC motors. However, DC motors are
still used for several reasons, including, wide speed
range, starting and accelerating torques more than
400% of their rated values, good speed regulation,
and simpler and cheaper control systems. Their
main applications include the manufacture of pulp,
paper, and paperboard, propulsion of electric
vehicles, textile industries, and public
transportation, such as subway and trolley systems.
Modern DC motor drives utilize power electronic
devices and are subdivided into chopper-fed and
controlled thyristor-fed drives, [1]. The
classification of DC motor drives can be done
according to the type of the utilized converter,
which controls the speed and the torque of the DC
motor, [2]. Nowadays, controlling DC motors with
high performance requires the use of DC-to-DC
converters, [3], [4], [5], [6], [7], [8], [9], [10], [11],
[12], [13], [14].
2 Four Quadrant Converter
The developed model for the two-quadrant
converter can be used as a building block in
developing the model for the four-quadrant
converter, [15]. As illustrated in Figure 1, the four-
quadrant converter is composed of two legs, A and
B, with each leg similar to the other.
Fig. 1: A developed model for a four-quadrant
converter
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Mustafa Saad, Khaled Mustafa
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The instantaneous voltage Va can be made either
equals.
For positive current
Va = Vdc when Q1 and Q2 are ON
Va = -Vdc when D3 and D4 are ON
Va = 0 when current freewheels through Q & D.
For negative current
Va = Vdc when D1 and D2 are ON
Va = -Vdc when Q3 and Q4 are ON
Va = 0 when current freewheels through Q &
D.
In the four-quadrant converter, two switching
schemes are normally employed which are the
bipolar switching scheme, and the unipolar
switching scheme, [15].
2.1 Bipolar Switching Scheme
Leg A and Leg B obtained the switching signals
from the same control signal. This implies that
switching Leg A and Leg B are always
complements. In a forward-breaking mode where
the average voltage Va is positive and smaller than
the back emf of the armature, the current will flow
through D1 and D2 when Va = Vdc and will flow
through Q3 and Q4 when Va = -Vdc. By using the
comparison between the control signal and
triangular waveform as shown in Figure 2, the
waveform and were obtained as shown in Figure
3.
Fig. 2: The control signal, Vc, and triangular
waveform, Vtri in a bipolar switching scheme.
The waveforms and were obtained from the
following rules:
Fig. 3: Waveforms for and after a comparison
between Vc and Vtri
2.2 Unipolar Switching Scheme
In the unipolar switching scheme, the switching
signal for Leg B is obtained from the inverse of the
control signal for Leg A. Figure 4 illustrates the
unipolar switching scheme and Figure 5 shows the
resultant waveform and .
Fig. 4: The control signals Vc and triangular
waveform, Vtri in a unipolar switching scheme
Fig. 5: The resultant waveform in a unipolar
switching scheme
The waveforms and were obtained from the
following rules:
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The relationship between Va(s) and Vc(s) for
both Bipolar and Unipolar is
3 Three-Phase Controlled Rectifier
The steady state and dynamic behavior of the
controlled rectifier are highly nonlinear which can
be described by a nonlinear differential equation.
Thus, to simplify the controller design for the
controlled rectifier, an average or linearized model
is used. However, tile approximation is only valid if
the bandwidth of the control loop is much lower
than the sampling frequency to ensure a continuous
current mode. Figure 6 shows the closed loop AC-
DC controlled rectifier.
Fig. 6: Close loop AC-DC controlled rectifier
The relationship between the average voltage and
the firing angle of a 3-phase controlled rectifier is
given by [16]
, f=50Hz and with =0.
=199.4
=190.416
4 Proportional Integral Controller
The proportional term and the integral term are
combined to form the PI controller. The PI
controller has a beneficiary effect on the steady-
state error since ti increases the system type of the
system by one. In general, the proportional term
affects the system’s stability. Where too high
proportional gain gives oscillation or an unstable
system. The integral term is used to eliminate the
steady-state error. However, setting the integral gain
too high is to invite oscillation, instability, and
integrator windup or actuator saturation. The
transfer function of the PI controller is
5 Simulation Results and Discussions
In this section, the simulation is done using Matlab
Simulink to see the characteristics of the two current
controller converters. Firstly, the 4-quadratic
unipolar converter has been done, then the 3-phase
controlled rectifier. Both current controllers were
simulated using the linear small signal model and
large signal model. Table 1 shows the system
parameters.
Table 1. System parameters
Component
Value
Vdc
200 v
Vtri
5 v
ftri
5000 Hz
Nrated
1313 rpm
Vrated
120 v
Ra
0.8
La
103 mH
KT = KE
0.764 vs/rad
J
2 Kg.m2
B
0.06 N.m.s
f
50 Hz
5.1 Simulation Result using 4-Quadrant
Converter
Use the parameters in Table 1 to design a small
signal model, which is shown in Figure 7. In this
research, the PID controller simulation block is used
and to use the PI controller the derivative term is set
to zero. Figure 8, and Figure 9 shows the Bode plot
and Pole Zero location of the open loop gain with
Kp=1 and Ki=0.
Fig. 7: Small signal model for 4-quadratic converter
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Fig. 8: Bode plot of a small signal model for a 4-
quadratic converter
Fig. 9: Pole zero location of a small signal model
From Figure 9, the system has two poles at 0.415
and 7.38. If the zero crossover frequency of the PI
controller is set at 4 rad/sec, then Ki /KP=4 rad/sec.
and if the DC gain is set to unity (by setting Ki =1),
then KP must be set to 0.25, the Bode plot for this
condition is shown in Figure 10.
Fig. 10: Open loop gain with Ki =1 and Kp =0.25
(solid line) for 4-quadratic converter
From Figure 10 it can be seen that the crossover
frequency or the bandwidth is too low. To increase
the bandwidth while maintaining the zero at 4
rad/sec, the value of needs to be increased. At the
same time, needs to be changed accordingly so
that the zero at 4 rad/sec is maintained. Since the
bandwidth is 500 Hz = 3.14103 rad/sec. the
magnitude curve has to be raised by 32.5 dB.
The new controller gains and are used in
the simulation diagram and the Bode plot is
obtained as shown in Figure 11.
Fig. 11: Bode plot for Ki = 42.169 & Kp = 10.542
(dash-dotted)
From Figure 11, the magnitude is nearly zero dB
and the phase margin is greater than 65, then we
take the computed values of Ki and Kp to be
implemented into the large signal simulation as in
Figure 12.
Bode Diagram
Frequency (rad/sec)
10-3 10-2 10-1 100101102103104
-90
-45
0
45
90
Phase (deg)
-40
-20
0
20
40
System: Model
I/O: small_model_q1/In1 (pt. 1) to DC-Motor Drive (pt. 2)
Frequency (rad/sec): 3.14e+003
Magnitude (dB): -20.5
From: smallmodelq1/In1 (pt. 1) To: DC-Motor Drive (pt. 2)
Magnitude (dB)
Pole-Zero Map
Real Axis
Imaginary Axis
-8 -7 -6 -5 -4 -3 -2 -1 0
-1
-0.5
0
0.5
1
System: Model (2)
Pole : -7.38
Damping: 1
Overshoot (%): 0
Frequency (rad/sec): 7.38
System: Model (2)
Pole : -0.415
Damping: 1
Overshoot (%): 0
Frequency (rad/sec): 0.415
System: Model (2)
Zero : -0.03
Damping: 1
Overshoot (%): 0
Frequency (rad/sec): 0.03
Bode Diagram
Frequency (rad/sec)
10-3 10-2 10-1 100101102103
-90
-45
0
45
90
Phase (deg)
-40
-20
0
20
40
60
80
System: Model (2)
I/O: ass1q1/In1 (pt. 1) to DC-Motor Drive (pt. 2)
Frequency (rad/sec): 3.14e+003
Magnitude (dB): -32.5
From: ass1q1/In1 (pt. 1) To: DC-Motor Drive (pt. 2)
Magnitude (dB)
Frequency (rad/sec)
10-3 10-2 10-1 100101102103
-90
-45
0
45
90
Phase (deg)
0
50
100
From: ass1q1/In1 (pt. 1) To: DC-Motor Drive (pt. 2)
Magnitude (dB)
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Fig. 12: Large signal model for closed-loop current control for 4-quadratic converter
5.2 Simulation Result using 3-Phase
Controlled Rectifier
The linear small signal model for the 3-phase
controlled rectifier is shown in Figure 13.
Fig. 13: Small signal model for 3-phase
converter
It is desired that the bandwidth of the current
controlled converter be 30 Hz. The poles are at the
same location because we use the same system so
we use the same pole-zero location as in Figure 9.
Therefore, Figure 14 shows the Bode plot of the
open loop gain with Kp=1 and Ki=0.
Fig. 14: Bode plot of a small signal model for a 3-
phase converter
Same as 4-quadratic. Ki =1 and Kp =0.25. The
Bode plot for this condition is shown in Figure 15.
Fig. 15: Open loop gain with Ki =1 and Kp =0.25
(solid line) for 3-phase converter
From Figure 15 it can be seen that the crossover
frequency or the bandwidth is too high. To decrease
the bandwidth while maintaining the zero at 4
rad/sec, the value of Ki needs to be decreased. At the
same time, Ki needs to be changed accordingly so
that the zero at 4 rad/sec is maintained. Since the
bandwidth is 30 Hz = 188.49 rad/sec, we need to
decrease the gain by 5.47 dB.
By putting the new values of and in the
simulation diagram, then the Bode plot of the new
values is shown in Figure 16.
From Figure 16, the magnitude is nearly zero dB
and the phase margin is greater than 65, then we
take the computed values of Ki and Kp to be
implemented into the large signal simulation as in
Figure 17.
Bode Diagram
Frequency (rad/sec)
10-3 10-2 10-1 100101102103
-90
-45
0
45
90
Phase (deg)
-20
0
20
40
60
80
100
System: Model (4)
I/O: small_model_q2/In1 (pt. 1) to DC-Motor Drive (pt. 2)
Frequency (rad/sec): 188
Magnitude (dB): 5.47
From: smallmodelq2/In1 (pt. 1) To: DC-Motor Drive (pt. 2)
Magnitude (dB)
Bode Diagram
Frequency (rad/sec)
10-3 10-2 10-1 100101102103
-90
-45
0
45
90
Phase (deg)
0
10
20
30
40
50
System: Model (2)
I/O: small_model_q2/In1 (pt. 1) to DC-Motor Drive (pt. 2)
Frequency (rad/sec): 188
Magnitude (dB): 17.5
From: smallmodelq2/In1 (pt. 1) To: DC-Motor Drive (pt. 2)
Magnitude (dB)
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Fig. 16: Bode plot for Ki = 0.5327 & Kp = 0.1331 (dash-dotted) for 3-phase converter
Fig. 17: Large signal model for closed-loop current control for 3-phase converter
Figure 18 and Figure 19 show the torque
response for the 4-quadratic converter and 3-phase
converter respectively.
Fig. 18: Torque response for 4-quadratic converter
Fig. 19: Torque response for 3-phase converter
Bode Diagram
Frequency (rad/sec)
10-3 10-2 10-1 100101102103
-90
-45
0
45
90
Phase (deg)
-20
0
20
40
60
80
100
From: smallmodelq2/In1 (pt. 1) To: DC-Motor Drive (pt. 2)
Magnitude (dB)
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14
0 0.2 0.4 0.6 0.8 1
0
5
10
15
20
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It can be seen from the simulation results that the
3-phase converter produces more ripples compared
to that of the switch-mode 4-quadratic converter,
which is almost not observable.
6 Conclusion
In this research, two different control models were
designed by using a 4-quadrant DC-DC converter
and a 3-phase rectifier to control the DC motor drive
in Matlab Simulink software. The PI controller
parameters have been designed based on the open
loop frequency response technique using a small
signal model, then used in a large signal model.
From the simulation results, it can be concluded that
the 4-quadratic converter has a faster response with
zero overshoot and zero steady-state error. In
addition, it tracked the reference signal almost
immediately. While the 3-phase controlled rectifier
has low control bandwidth and high-frequency
current ripple.
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Mustafa Saad, Khaled Mustafa
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Mustafa Saad carried out the simulation and the
controller design.
- Mustafa Saad and Khaled Mustafa have organized
and executed the research of Section 5.
- Khaled Mustafa was responsible for the text
writing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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