Control of Direct Current Machine by the Change of Resistance in Armature Circuit
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The paper discusses the control of DC motor rotation speed through resistance changes in the armature circuit, specifically focusing on resistor control methods. Various techniques for regulating armature resistance are analyzed, including discrete shunting of resistors using contactors and electronic keys to manage speed more efficiently. The study also emphasizes the design of control systems, highlighting the nonlinear dynamic behavior of DC machines and offering recommendations for integral and proportional regulators in current and speed control loops.
![International Journal of Innovation Engineering and Science Research
Open Access
Volume 2 Issue 4 July-August 2018 51|P a g e
ABSTRACT
–
Control of Direct Current Machine by the Change
of Resistance in Armature Circuit
BIYA MOTTO Frederic1k
, TCHUIDJAN Roger2
, NDZANA Benoît2
, TATSA
TCHINDA COLINCE3
.
1- Faculty of Science, University of Yaounde I, PO.Box 812 Yaoundé, Cameroun;
2- National Advanced School of Engineering of Yaounde;
3- Mekin Hydroelectric Development Corporation, PO.Box 13155 Yaoundé, Cameroon;
The control of motor rotation speed by the change of resistor resistance value in armature circuit is
called ‘resistor control”. For the regulation of resistance value R0, included in armature winding circuit,
we can use various technical solutions. The most used solution is the discrete variation of armature
added resistance value by shunting its parts with contactors contacts.
Nowadays, the change of resistor resistance in armature circuit can be realized by shunting
with a given porosity γ of resistor R0 trough electronic keys. In this paper, we study the design of
control system represented on figure 1.
Keywords: Control of DC machine, change of resistance, armature circuit
I. INTRODUCTION
DC motors consist of rotor-mounted windings (armature) and stationary windings (field poles). In all
DC motors, except permanent magnet brushless motors, current must be conducted to the armature
windings by passing current through carbon brushes that slide over a set of copper surfaces called a
commutator, which is mounted on the rotor.
[1][2]
The commutator bars are soldered to armature coils. The brush/commutator combination
makes a sliding switch that energizes particular portions of the armature, based on the position of the
rotor. This process creates north and south magnetic poles on the rotor that are attracted to or
repelled by south and north poles on the stator, magnetic attraction and repulsion that causes the
rotor to rotate.[3][4]
The dynamic behavior of DC machine is mainly determined by the type of the connection
between the excitation winding and the armature winding including the commutation and
compensation winding.
The greatest advantage of DC motors may be speed control. Since speed is directly
proportional to armature voltage and inversely proportional to the magnetic flux produced by the
poles, adjusting the armature voltage or the field current will change the rotor speed.
[5]
Speed control means change of a speed to a value required for performing the specific work
process. This adjustment should not be taken to include the natural change in speed which occurs
due to the change in the load on the drive shaft. The electrical speed control has many economical as
well as engineering advantages over mechanical speed control. There are so many methods for
controlling the speed of a DC shunt motor but field rheostat control method is most reliable, economic
and independent of load on the motor. This method is only applicable when we want speed which is
higher than the normal speed of the motor. In this method, an increase in controlling resistance
reduces the field current with a consequent reduction in flux and an increase in speed. But if we want](https://image.slidesharecdn.com/9114259953-181225111618/85/Control-of-Direct-Current-Machine-by-the-Change-of-Resistance-in-Armature-Circuit-1-320.jpg)
![BIYA MOTTO Frederic et al. “International Journal of Innovation Engineering and Science Research”
Volume 2 Issue 4 July-August 2018 52|P a g e
to obtain low speed to control the low speed mechanical drive, we use armature rheostat control
method. In this method, the speed at full load can be reduced to any desired value depending on the
amount of resistance. But if we use both techniques in same machine then we can control motor from
zero speed to maximum.[6]
In field control the adjustment can be obtained by means of a small
rheostat and relatively good speed regulation is obtained for all speed but with the armature control a
bulky resistance is required. So if we use both methods simultaneously, cost of the machine will
increase a little but we will get a large range of speed control. To neutralize the effect of power loss
heat sink can be used. So by this method we can control the speed of a DC shunt motor to perform
various tasks in effective and economic way.
II. EQUATIONS OF ELECTRIC DRIVE POWER CHANNEL WITH REGULATION OF
RESISTANCE IN ARMATURE CIRCUIT
The system of resistor control of electromotor M rotation speed with separate excitation is
composed of additive resistance R0 in armature circuit, transistor VT, transistor control system CS VT,
current captor CC1 with shunt RS1, speed captor BR and control installation (figure 1)
We shall assume that we supply in excitation winding and in armature winding direct current nominal
voltage UN. Additive resistor R0 and transistor VT with control system CS VT constitute electrical
transducer. The control of resistance value in armature circuit is done by the switch of transistor VT
with porosity γ є [0,1].
If the working period of transistor control pulses is much more higher than the time constant of
armature circuit, then we can prove that the equivalent resistance R included in armature circuit is
𝑅 = 𝛾𝑅0
Thus the input signal of electrical transducer is the porosity γ є [0,1], and the output is the
equivalent resistance R. By varying R, we change armature current 𝑖2 and the electromagnetic torque
(moment) M.
To the electric drive resistor circuit in figure 1, we can match an armature equivalent circuit
represented in figure 2.
Figure 2: Armature circuit equivalent circuit](https://image.slidesharecdn.com/9114259953-181225111618/85/Control-of-Direct-Current-Machine-by-the-Change-of-Resistance-in-Armature-Circuit-2-320.jpg)
![BIYA MOTTO Frederic et al. “International Journal of Innovation Engineering and Science Research”
Volume 2 Issue 4 July-August 2018 53|P a g e
In that circuit, R2 – active resistance of armature;
L02 – armature inductance;
𝛾𝑅0- Additive resistance; γ є [0,1];
𝐸 = 𝛹 - armature e.m.f
𝛹 = 𝑈 𝑁/ 𝐵 – constant expression;
𝐵 – Basic armature rotation speed.
The state variables control system that describes the DC machine dynamic properties will
look as follows:
𝑅2
∗
. 𝑇2 𝑝𝑖2
∗
+ 𝑖2
∗
= 1 − 𝜔∗
− 𝛾𝑅0
∗
. 𝑖2
∗
;
𝑇 𝑀𝑒𝑐 ℎ . 𝑝𝜔∗
= 𝑖2
∗
− 𝐼𝑟
∗ (1)
We have 𝑖2
∗
= 𝑀∗
, 𝐼𝑟
∗
= 𝑀𝑐
∗
The equation (1) is nonlinear because of the presence of expression𝛾𝑅0
∗
. 𝑖2
∗
.
The equation (1) corresponds to the structural circuit shown on figure 3. That system has two
input signals: pulses porosity γ and current 𝐼𝑟
∗
.
γis the control signal while 𝐼𝑟
∗
is the perturbation.
III. DESIGN OF CONTROL INSTALLATION
The resistance control system is constructed according to subordinate principle. It is
composed of internal armature current loop and external speed loop. The current loop forms the
control signal γ.
According to equations (1), established armature current value is
𝑖2
∗
=
1 − 𝜔∗
𝑅2
∗
+ 𝛾𝑅0
∗
From the last expression, when the porosity value γ increases the armature current 𝑖2
∗
decreases. Let us introduce new control variable𝑥, linked with γ. We consider𝑥 = 1
𝛾.
Therefore with the increase of𝑥, the armature current 𝑖2
∗
will also increase.
The structural circuit of subordinate control system is shown on figure 3. The control system is
composed of internal current loop with integral regulator. The transfer coefficient of current captor 𝐾𝑐
∗
is found from the condition 𝐾𝑐
∗
. 𝐼 𝑚𝑎𝑥
∗
= 1, thus 𝐾𝑐
∗
= 1/𝐼 𝑚𝑎𝑥
∗
𝑝 At the entrance of current loop, we install a current limit element. The maximal current value
𝐼 𝑚𝑎𝑥
∗
should be limited to ensure given static and dynamic loads in electric drive mechanism and
reliable functioning of collector-mechanism. As a rule, the armature maximal current value 𝐼 𝑚𝑎𝑥
∗
is
equal to 1,2;….;2,0.
The current limitation at a given level 𝐼 𝑚𝑎𝑥
∗
can be achieved from the limitation of current loop
input signal 𝑥1
∗
by the value𝑥1𝑚𝑎𝑥
∗
. If we are given the armature current loop maximal value 𝐼 𝑚𝑎𝑥
∗
, then
𝑥1 𝑚𝑎𝑥
∗
= 𝐾𝑐
∗
. 𝐼 𝑚𝑎𝑥
∗
= 1](https://image.slidesharecdn.com/9114259953-181225111618/85/Control-of-Direct-Current-Machine-by-the-Change-of-Resistance-in-Armature-Circuit-3-320.jpg)
![BIYA MOTTO Frederic et al. “International Journal of Innovation Engineering and Science Research”
Volume 2 Issue 4 July-August 2018 54|P a g e
IV. DEFINITION OF CURRENT AND SPEED LOOPS PARAMETERS
The parameter of integral current regulator 𝑇𝑟2
should be such that the transient processes in
current control loop will have an etalon aspect. To the current loop corresponds the system of
differential equations:
𝑇𝑟2
. 𝑝𝑥∗
= 𝑥1
∗
− 𝐾𝑐
∗
. 𝑖2
∗
;
𝑅2
∗
. 𝑇2. 𝑝𝑖2
∗
+ 𝑖2
∗
+ 𝑅0
∗
.
𝑖2
∗
𝑥∗
= 1 − 𝜔∗
That system is nonlinear on control signal 𝑥∗
. For the determination of time constant for
integral regulator Tr2, we linearize those equations in neighborhood of the working point [𝑥 ∞ ∗
,
𝑖2 ∞ ∗
]
We observe that the mechanical time constant 𝑇 𝑀𝑒𝑐 ℎis sensitively higher than the armature
time constant 𝑇2and the speed 𝜔∗
practically does not change with current regulation, ie ∆𝜔∗
= 0.
Considering p=0, 𝑥∗
= 𝑥(∞)∗
, 𝑖2
∗
= 𝑖2(∞)∗
,
We have equations of stationary working regime:
0 = 𝑥1(∞)∗
− 𝐾𝑐
∗
. 𝑖2 ∞ ∗
;
𝑅2
∗
. 𝑖2(∞)∗
+
𝑅0
∗
. 𝑖2(∞)∗
𝑥(∞)∗
= 1 − 𝜔∗
The solution of that system defines the stationary values of variables
𝑥(∞)∗
=
𝑅0
∗
. 𝑖2(∞)∗
1 − 𝜔∗ − 𝑅2
∗
. 𝑖2(∞)∗
; 𝑖2(∞)∗
=
𝑥1(∞)∗
𝐾𝑐
∗
From linearization results we have the system of equations in variations:
𝑇𝑟2
. 𝑝∆𝑥∗
= ∆𝑥1
∗
− 𝐾𝑐
∗
. ∆𝑖2
∗
;
𝑅2
∗
. 𝑇2. 𝑝∆𝑖2
∗
+ ∆𝑖2
∗
+
𝑅0
∗
𝑥(∞)∗ . ∆𝑖2
∗
−
𝑅0
∗.𝑖2 ∞ ∗
𝑥 ∞ ∗ ∆𝑥∗
= 0
(2)
From the linearized equations system, we find the transfer function of current loop:
𝑊𝑐𝑙 =
∆𝑖2
∗
∆𝑥1
∗ =
1 𝐾𝑐
∗
𝑇𝜇
2 𝑝2 + 𝑑. 𝑇𝜇 𝑝 + 1
Where 𝑇𝜇 = 𝑑. 𝑇2
𝑅2
∗.𝑖2(∞)∗
1−𝜔∗ ;
𝑑 =
1 − 𝜔∗
1 − 𝜔∗ − 𝑅2
∗
. 𝑖2(∞)∗
.
𝑅0
∗
. 𝑇𝑟2
𝑥1(∞)∗. 𝑅2
∗
. 𝑇2
≈
𝑅0
∗
. 𝑇𝑟2
𝑥1(∞)∗. 𝑅2
∗
. 𝑇2
The damping factor d of transient function will have the minimal value for 𝑥1(∞)∗
= 𝐾𝑐
∗
. 𝐼 𝑚𝑎𝑥
∗
= 1:
𝑑 𝑚𝑖𝑛 =
𝑅0
∗
. 𝑇𝑟2
𝑅2
∗
. 𝑇2
If we consider 𝑑 𝑚𝑖𝑛 = 2, then the current regulator time constant](https://image.slidesharecdn.com/9114259953-181225111618/85/Control-of-Direct-Current-Machine-by-the-Change-of-Resistance-in-Armature-Circuit-4-320.jpg)
![BIYA MOTTO Frederic et al. “International Journal of Innovation Engineering and Science Research”
Volume 2 Issue 4 July-August 2018 55|P a g e
𝑇𝑟2 = 2. 𝑇2. 𝑅2
∗
/𝑅0
∗
.
The current loop time constant 𝑇𝜇 will considerably depend on the drive working regime. For
𝜔∗
= 1 − 𝑅2
∗
. 𝑖2(∞)∗
, the time constant will have the highest value 𝑇𝜇 = 2. 𝑇2.
If we assume that the current loop has transfer function
𝑊𝑐𝑙 =
𝑖2
∗
𝑥1
∗ =
1 𝐾 𝑐
∗
2.𝑇2 𝑝2+1
,
then the speed regulator transfer function is on technical optimum
𝑊𝑠𝑟 =
2. 𝑇2 𝑝 + 1 𝐾𝑐
∗
. 𝑇 𝑀𝑒𝑐 ℎ . 𝑃
2𝑇𝜇2 𝑃. 𝑇𝜇2 𝑃 + 1
=
𝐾𝑐
∗
. 𝑇 𝑀𝑒𝑐 ℎ
2𝑇𝑟2
= 𝐾𝑠𝑟
∗
,
where 𝑇𝜇2 = 2. 𝑇2.
The set of electromechanical (mechanical) characteristics of closed loop control system with
speed regulator will look as shown on figure 5.
Figure 5: The set of electromechanical (mechanical) characteristics with closed loop control system
V. CONCLUSIONS
The dynamic model of Direct Current electrical drive with control by the means of armature
circuit resistance change is nonlinear on control signal. For the construction of current control loop,
the design of regulators is done with linearization of equations (2). It is recommended to use integral
regulator in current loop. At the input of current control loop, we install an armature current limitator.
The speed control loop is external compared to current control loop and it should have a proportional
regulator.
The region of speed regulation by commutation of resistor in armature circuit for the electric
drive is limited by the value of resistor resistance R0 (figure 5). The domain enlargement of speed
regulation by reduction of expression R0 will lead to high commutation survoltages in the transistor
and its destruction.
The enlargement of speed regulation domain can be reached by in series switching of
resistors, with transistor shunts. The reduction of porosity on transistor functioning should be done in
opposite manner.
A deep resistor speed regulation is not energically efficient, because of energy losses.
REFERENCES
[1] www.google.com/speedcontrolofshuntmotor
[2] www.wikipedia.com/shuntmotor
[3] Theory and performance of electrical machine by J.B.Gubta
[4] Electrical machine by I.J Nagrath and D.P. Kothari
[5] Electrical machine by P.S. Bhimra
[6] www.electrical4u.com](https://image.slidesharecdn.com/9114259953-181225111618/85/Control-of-Direct-Current-Machine-by-the-Change-of-Resistance-in-Armature-Circuit-5-320.jpg)
Control of Direct Current Machine by the Change of Resistance in Armature Circuit
- 1. International Journal of Innovation Engineering and Science Research Open Access Volume 2 Issue 4 July-August 2018 51|P a g e ABSTRACT – Control of Direct Current Machine by the Change of Resistance in Armature Circuit BIYA MOTTO Frederic1k , TCHUIDJAN Roger2 , NDZANA Benoît2 , TATSA TCHINDA COLINCE3 . 1- Faculty of Science, University of Yaounde I, PO.Box 812 Yaoundé, Cameroun; 2- National Advanced School of Engineering of Yaounde; 3- Mekin Hydroelectric Development Corporation, PO.Box 13155 Yaoundé, Cameroon; The control of motor rotation speed by the change of resistor resistance value in armature circuit is called ‘resistor control”. For the regulation of resistance value R0, included in armature winding circuit, we can use various technical solutions. The most used solution is the discrete variation of armature added resistance value by shunting its parts with contactors contacts. Nowadays, the change of resistor resistance in armature circuit can be realized by shunting with a given porosity γ of resistor R0 trough electronic keys. In this paper, we study the design of control system represented on figure 1. Keywords: Control of DC machine, change of resistance, armature circuit I. INTRODUCTION DC motors consist of rotor-mounted windings (armature) and stationary windings (field poles). In all DC motors, except permanent magnet brushless motors, current must be conducted to the armature windings by passing current through carbon brushes that slide over a set of copper surfaces called a commutator, which is mounted on the rotor. [1][2] The commutator bars are soldered to armature coils. The brush/commutator combination makes a sliding switch that energizes particular portions of the armature, based on the position of the rotor. This process creates north and south magnetic poles on the rotor that are attracted to or repelled by south and north poles on the stator, magnetic attraction and repulsion that causes the rotor to rotate.[3][4] The dynamic behavior of DC machine is mainly determined by the type of the connection between the excitation winding and the armature winding including the commutation and compensation winding. The greatest advantage of DC motors may be speed control. Since speed is directly proportional to armature voltage and inversely proportional to the magnetic flux produced by the poles, adjusting the armature voltage or the field current will change the rotor speed. [5] Speed control means change of a speed to a value required for performing the specific work process. This adjustment should not be taken to include the natural change in speed which occurs due to the change in the load on the drive shaft. The electrical speed control has many economical as well as engineering advantages over mechanical speed control. There are so many methods for controlling the speed of a DC shunt motor but field rheostat control method is most reliable, economic and independent of load on the motor. This method is only applicable when we want speed which is higher than the normal speed of the motor. In this method, an increase in controlling resistance reduces the field current with a consequent reduction in flux and an increase in speed. But if we want
- 2. BIYA MOTTO Frederic et al. “International Journal of Innovation Engineering and Science Research” Volume 2 Issue 4 July-August 2018 52|P a g e to obtain low speed to control the low speed mechanical drive, we use armature rheostat control method. In this method, the speed at full load can be reduced to any desired value depending on the amount of resistance. But if we use both techniques in same machine then we can control motor from zero speed to maximum.[6] In field control the adjustment can be obtained by means of a small rheostat and relatively good speed regulation is obtained for all speed but with the armature control a bulky resistance is required. So if we use both methods simultaneously, cost of the machine will increase a little but we will get a large range of speed control. To neutralize the effect of power loss heat sink can be used. So by this method we can control the speed of a DC shunt motor to perform various tasks in effective and economic way. II. EQUATIONS OF ELECTRIC DRIVE POWER CHANNEL WITH REGULATION OF RESISTANCE IN ARMATURE CIRCUIT The system of resistor control of electromotor M rotation speed with separate excitation is composed of additive resistance R0 in armature circuit, transistor VT, transistor control system CS VT, current captor CC1 with shunt RS1, speed captor BR and control installation (figure 1) We shall assume that we supply in excitation winding and in armature winding direct current nominal voltage UN. Additive resistor R0 and transistor VT with control system CS VT constitute electrical transducer. The control of resistance value in armature circuit is done by the switch of transistor VT with porosity γ є [0,1]. If the working period of transistor control pulses is much more higher than the time constant of armature circuit, then we can prove that the equivalent resistance R included in armature circuit is 𝑅 = 𝛾𝑅0 Thus the input signal of electrical transducer is the porosity γ є [0,1], and the output is the equivalent resistance R. By varying R, we change armature current 𝑖2 and the electromagnetic torque (moment) M. To the electric drive resistor circuit in figure 1, we can match an armature equivalent circuit represented in figure 2. Figure 2: Armature circuit equivalent circuit
- 3. BIYA MOTTO Frederic et al. “International Journal of Innovation Engineering and Science Research” Volume 2 Issue 4 July-August 2018 53|P a g e In that circuit, R2 – active resistance of armature; L02 – armature inductance; 𝛾𝑅0- Additive resistance; γ є [0,1]; 𝐸 = 𝛹 - armature e.m.f 𝛹 = 𝑈 𝑁/ 𝐵 – constant expression; 𝐵 – Basic armature rotation speed. The state variables control system that describes the DC machine dynamic properties will look as follows: 𝑅2 ∗ . 𝑇2 𝑝𝑖2 ∗ + 𝑖2 ∗ = 1 − 𝜔∗ − 𝛾𝑅0 ∗ . 𝑖2 ∗ ; 𝑇 𝑀𝑒𝑐 ℎ . 𝑝𝜔∗ = 𝑖2 ∗ − 𝐼𝑟 ∗ (1) We have 𝑖2 ∗ = 𝑀∗ , 𝐼𝑟 ∗ = 𝑀𝑐 ∗ The equation (1) is nonlinear because of the presence of expression𝛾𝑅0 ∗ . 𝑖2 ∗ . The equation (1) corresponds to the structural circuit shown on figure 3. That system has two input signals: pulses porosity γ and current 𝐼𝑟 ∗ . γis the control signal while 𝐼𝑟 ∗ is the perturbation. III. DESIGN OF CONTROL INSTALLATION The resistance control system is constructed according to subordinate principle. It is composed of internal armature current loop and external speed loop. The current loop forms the control signal γ. According to equations (1), established armature current value is 𝑖2 ∗ = 1 − 𝜔∗ 𝑅2 ∗ + 𝛾𝑅0 ∗ From the last expression, when the porosity value γ increases the armature current 𝑖2 ∗ decreases. Let us introduce new control variable𝑥, linked with γ. We consider𝑥 = 1 𝛾. Therefore with the increase of𝑥, the armature current 𝑖2 ∗ will also increase. The structural circuit of subordinate control system is shown on figure 3. The control system is composed of internal current loop with integral regulator. The transfer coefficient of current captor 𝐾𝑐 ∗ is found from the condition 𝐾𝑐 ∗ . 𝐼 𝑚𝑎𝑥 ∗ = 1, thus 𝐾𝑐 ∗ = 1/𝐼 𝑚𝑎𝑥 ∗ 𝑝 At the entrance of current loop, we install a current limit element. The maximal current value 𝐼 𝑚𝑎𝑥 ∗ should be limited to ensure given static and dynamic loads in electric drive mechanism and reliable functioning of collector-mechanism. As a rule, the armature maximal current value 𝐼 𝑚𝑎𝑥 ∗ is equal to 1,2;….;2,0. The current limitation at a given level 𝐼 𝑚𝑎𝑥 ∗ can be achieved from the limitation of current loop input signal 𝑥1 ∗ by the value𝑥1𝑚𝑎𝑥 ∗ . If we are given the armature current loop maximal value 𝐼 𝑚𝑎𝑥 ∗ , then 𝑥1 𝑚𝑎𝑥 ∗ = 𝐾𝑐 ∗ . 𝐼 𝑚𝑎𝑥 ∗ = 1
- 4. BIYA MOTTO Frederic et al. “International Journal of Innovation Engineering and Science Research” Volume 2 Issue 4 July-August 2018 54|P a g e IV. DEFINITION OF CURRENT AND SPEED LOOPS PARAMETERS The parameter of integral current regulator 𝑇𝑟2 should be such that the transient processes in current control loop will have an etalon aspect. To the current loop corresponds the system of differential equations: 𝑇𝑟2 . 𝑝𝑥∗ = 𝑥1 ∗ − 𝐾𝑐 ∗ . 𝑖2 ∗ ; 𝑅2 ∗ . 𝑇2. 𝑝𝑖2 ∗ + 𝑖2 ∗ + 𝑅0 ∗ . 𝑖2 ∗ 𝑥∗ = 1 − 𝜔∗ That system is nonlinear on control signal 𝑥∗ . For the determination of time constant for integral regulator Tr2, we linearize those equations in neighborhood of the working point [𝑥 ∞ ∗ , 𝑖2 ∞ ∗ ] We observe that the mechanical time constant 𝑇 𝑀𝑒𝑐 ℎis sensitively higher than the armature time constant 𝑇2and the speed 𝜔∗ practically does not change with current regulation, ie ∆𝜔∗ = 0. Considering p=0, 𝑥∗ = 𝑥(∞)∗ , 𝑖2 ∗ = 𝑖2(∞)∗ , We have equations of stationary working regime: 0 = 𝑥1(∞)∗ − 𝐾𝑐 ∗ . 𝑖2 ∞ ∗ ; 𝑅2 ∗ . 𝑖2(∞)∗ + 𝑅0 ∗ . 𝑖2(∞)∗ 𝑥(∞)∗ = 1 − 𝜔∗ The solution of that system defines the stationary values of variables 𝑥(∞)∗ = 𝑅0 ∗ . 𝑖2(∞)∗ 1 − 𝜔∗ − 𝑅2 ∗ . 𝑖2(∞)∗ ; 𝑖2(∞)∗ = 𝑥1(∞)∗ 𝐾𝑐 ∗ From linearization results we have the system of equations in variations: 𝑇𝑟2 . 𝑝∆𝑥∗ = ∆𝑥1 ∗ − 𝐾𝑐 ∗ . ∆𝑖2 ∗ ; 𝑅2 ∗ . 𝑇2. 𝑝∆𝑖2 ∗ + ∆𝑖2 ∗ + 𝑅0 ∗ 𝑥(∞)∗ . ∆𝑖2 ∗ − 𝑅0 ∗.𝑖2 ∞ ∗ 𝑥 ∞ ∗ ∆𝑥∗ = 0 (2) From the linearized equations system, we find the transfer function of current loop: 𝑊𝑐𝑙 = ∆𝑖2 ∗ ∆𝑥1 ∗ = 1 𝐾𝑐 ∗ 𝑇𝜇 2 𝑝2 + 𝑑. 𝑇𝜇 𝑝 + 1 Where 𝑇𝜇 = 𝑑. 𝑇2 𝑅2 ∗.𝑖2(∞)∗ 1−𝜔∗ ; 𝑑 = 1 − 𝜔∗ 1 − 𝜔∗ − 𝑅2 ∗ . 𝑖2(∞)∗ . 𝑅0 ∗ . 𝑇𝑟2 𝑥1(∞)∗. 𝑅2 ∗ . 𝑇2 ≈ 𝑅0 ∗ . 𝑇𝑟2 𝑥1(∞)∗. 𝑅2 ∗ . 𝑇2 The damping factor d of transient function will have the minimal value for 𝑥1(∞)∗ = 𝐾𝑐 ∗ . 𝐼 𝑚𝑎𝑥 ∗ = 1: 𝑑 𝑚𝑖𝑛 = 𝑅0 ∗ . 𝑇𝑟2 𝑅2 ∗ . 𝑇2 If we consider 𝑑 𝑚𝑖𝑛 = 2, then the current regulator time constant
- 5. BIYA MOTTO Frederic et al. “International Journal of Innovation Engineering and Science Research” Volume 2 Issue 4 July-August 2018 55|P a g e 𝑇𝑟2 = 2. 𝑇2. 𝑅2 ∗ /𝑅0 ∗ . The current loop time constant 𝑇𝜇 will considerably depend on the drive working regime. For 𝜔∗ = 1 − 𝑅2 ∗ . 𝑖2(∞)∗ , the time constant will have the highest value 𝑇𝜇 = 2. 𝑇2. If we assume that the current loop has transfer function 𝑊𝑐𝑙 = 𝑖2 ∗ 𝑥1 ∗ = 1 𝐾 𝑐 ∗ 2.𝑇2 𝑝2+1 , then the speed regulator transfer function is on technical optimum 𝑊𝑠𝑟 = 2. 𝑇2 𝑝 + 1 𝐾𝑐 ∗ . 𝑇 𝑀𝑒𝑐 ℎ . 𝑃 2𝑇𝜇2 𝑃. 𝑇𝜇2 𝑃 + 1 = 𝐾𝑐 ∗ . 𝑇 𝑀𝑒𝑐 ℎ 2𝑇𝑟2 = 𝐾𝑠𝑟 ∗ , where 𝑇𝜇2 = 2. 𝑇2. The set of electromechanical (mechanical) characteristics of closed loop control system with speed regulator will look as shown on figure 5. Figure 5: The set of electromechanical (mechanical) characteristics with closed loop control system V. CONCLUSIONS The dynamic model of Direct Current electrical drive with control by the means of armature circuit resistance change is nonlinear on control signal. For the construction of current control loop, the design of regulators is done with linearization of equations (2). It is recommended to use integral regulator in current loop. At the input of current control loop, we install an armature current limitator. The speed control loop is external compared to current control loop and it should have a proportional regulator. The region of speed regulation by commutation of resistor in armature circuit for the electric drive is limited by the value of resistor resistance R0 (figure 5). The domain enlargement of speed regulation by reduction of expression R0 will lead to high commutation survoltages in the transistor and its destruction. The enlargement of speed regulation domain can be reached by in series switching of resistors, with transistor shunts. The reduction of porosity on transistor functioning should be done in opposite manner. A deep resistor speed regulation is not energically efficient, because of energy losses. REFERENCES [1] www.google.com/speedcontrolofshuntmotor [2] www.wikipedia.com/shuntmotor [3] Theory and performance of electrical machine by J.B.Gubta [4] Electrical machine by I.J Nagrath and D.P. Kothari [5] Electrical machine by P.S. Bhimra [6] www.electrical4u.com