S.C.

1

2011

Featuring 6 pulse Thyristor bridge and cascade control | Eng. Souad Chamieh

S.C. DC MOTOR – MODELING, CONTROL AND

SIMULATION ON MATLAB-SIMULINK

S.C.

2

S.C.

3

Table of Contents I- Introduction: .................................................................................................................. 4

II- Modeling of DC Motor .................................................................................................. 4

2.1 Modeling and Block diagram of DC motor. ................................................................. 4

2.1.1Operation at no-load: ............................................................................................. 4

2.1.2 Operation at load: ................................................................................................. 5

III- Simulation of the model on Matlab-Simulink.............................................................. 6

IV- Three phase fully controlled bridge rectifier - modeling on Matlab Simulink ............ 11

4.1 Summary featuring the principle of operation of a 3-phase thyristor bridge ................ 11

4.2 Schematic of a 3-phase thyristor bridge on Matlab-Simulink ..................................... 14

4.3 Command of the 3-phase Thyristor bridge on Matlab-Simulink ................................. 15

4.5 Simulation of the command – Thyristor bridge .......................................................... 16

V- Simulation of the complete system: DC motor and inverter. ........................................ 19

VI- Control System ......................................................................................................... 22

6.1 Cascade controller ..................................................................................................... 22

6.2 Design objectives ...................................................................................................... 24

6.3 Loop tuning steps ...................................................................................................... 24

6.4 Study of the current Loop .......................................................................................... 25

6.5 Study of the angular velocity Loop ............................................................................ 31

6.5.1 Block diagram, closed loop transfer function and applying a proportional gain ... 31

6.5.2 Applying a proportional - Integral (PI) controller ................................................ 32

6.5.2 Simulation results ............................................................................................... 34

6.5.2 Summary of final results ..................................................................................... 38

Appendix ............................................................................................................................ 40

Numerical application of the project: ............................................................................... 40

Matlab: ............................................................................................................................ 40

List of used subsystems: .................................................................................................. 40

S.C.

4

Ia .k Ce

.k E

max

max

I- Introduction:

This project aims to study the control of a separately-excited DC motor with variable speed.

The DC motor is powered through a controlled rectifier bridge.

We seek in this work to manage two parameters of the machine: the current consumption

which should not exceed in any case the maximum current supported by the machine as well

as the variable speed.

This project will have three main parts:

1- Modeling of the DC motor.

2- Modeling of the inverter (bridge rectifier with Thyristors)

3- System control

II- Modeling of DC Motor

2.1 Modeling and Block diagram of DC motor.

The speed control of this machine is achieved by manipulating the voltage applied at the

armature coils.

The field coils are separately excited by a constant current.

The excitation flux is considered constant and equal to the maximum flux.

Figure 2.1: Equivalent schematic of a separately excited DC motor

2.1.1Operation at no-load: The two fundamental equations of a DC motor are given by:

eq 2.1

eq 2.2

S.C.

5

dt

dILIREV a

aaaa

)()().()(

)(.)(

)(.)(

max

max

pEpIpLRpV

pIkpC

pkpE

aaaa

ae

dt

dJCCe tresistan

CrfC tresis .tan

E: back e.m.f [V]

k: electromotive force constant [Nm/A]

Ce: electromagnetic torque [Nm]

Ω : angular velocity [rad/s]

Ia: Armature current [A]

Armature voltage Va is given by (see fig 2.1):

eq 2.3

La: Armature inductance [H]

Ra: Armature resistance [Ω]

Va: Armature applied voltage [V]

Applying Laplace transform to the above equations:

eq 2.4

eq 2.5

eq 2.6

The Block diagram of the motor at no load is displayed in the following figure:

Figure 2.2: Block diagram at No Load (open Loop) – Matlab Simulink

2.1.2 Operation at load: The mechanical equation:

eq 2.7

With eq 2.8

f: friction

J Inertia of motor and load

S.C.

6

Cr: resistant Torque

Ce: electromagnetic torque

Ω : angular velocity

Using equations 2.4, 2.5, 2.6, 2.7 and 2.8 we get the closed loop block diagram of the DC

motor:

Figure 2.3: Block diagram of separately excited DC motor (closed Loop) – Matlab Simulink

III- Simulation of the model on Matlab-Simulink

In the simulation, the input voltage is an echelon or a step that increases from 0 to 260V.

Consider the following motor characteristics:

Nominal armature voltage: Un=260V

Nominal angular velocity: Nn=2150 tr/min= 225 rad/s

Armature inductance: La=34 mH

Armature resistance: Ra=1.26 Ω

Nominal Torque: Cn=14 N.m

Armature nominal current: Ian=13.5 A

Friction : f = 0.01

Inertia : J = 0.02 kg.m2

Figure 3.1: Block diagram of Figure 2.3 prepared for simulation on Matlab Simulink.

S.C.

7

In figure 3.1:

- Step1block is the step input armature Voltage (echelon input).

- Step block is the step input resistive torque (load torque)

- I: the armature current taken as an output for visualization.

- C: the torque which is proportional to the current by the gain Kmax.

- W is the output angular velocity at the rotor’s shaft.

Note: during simulation in Simulink, the outputs are stored in variables of type “table” in the

workplane of Matlab. The variables hold the same name as the output block, for example t for

the clock time.

Plotting the variables is done by simple Matlab commands, for example to plot the current in

function of time, simply type plot(t,I).

If after plot, the figures or waveforms are not clear or don’t make much sense then proceed as

follows:

- Check if simulation time is well adjusted. Simulation time is not set automatically and

needs to be adjusted manually. The default value of Matlab simulation time is usually

10s.

For example the simulation time in figure 3.2 is 2s. If it has been set to 100s, then the

starting curve will hardly be readable and the entire graph will appear as to be a

straight line similar to the line between 0.4s and 2s in figure 3.2.

- Check if sampling time or sampling type is also adjusted. Usually it’s better leaving it

to default or auto-adjust.

- Check if your system is well connected and studied. And make sure that the functions,

variables and constants are correctly inserted.

It is important to realize that it is up to the user to verify the reliability, suitability and

accuracy of the simulation. In other words, a good engineer anticipates simulation results.

Simulation Results:

Figure 3.2: Open-loop response at no-load operation, current in function of time.

S.C.

8

Figure 3.3: Open-loop response at no-load operation, Torque in function of time.

Figure 3.4: Open-loop response at no-load operation, angular velocity in function of time.

Notice that at No-load operation, the starting current is extremely high.

S.C.

9

Figure 3.5: closed-loop response - load operation, current in function of time.

Figure 3.6: closed-loop response - load operation, Torque in function of time.

S.C.

10

Figure 3.7: closed-loop response - load operation, angular velocity in function of time.

In the following, the model of the machine given by figure 2.3 or 3.1, will be simplified in a

subsystem shown in figure 3.8.

Inputs of this subsystem:

Cr: resistance torque (load torque).

Va: applied armature voltage.

Outputs :

Ce: electromagnetic torque.

Ia: armature current.

N: angular velocity

Figure 3.8: closed-loop response - load operation, angular velocity in function of time.

S.C.

11

IV- Three phase fully controlled bridge rectifier - modeling on Matlab Simulink

4.1 Summary featuring the principle of operation of a 3-phase thyristor bridge

Figure 4.1: schematic of a 3-phase thyristor bridge.

Figure 4.2: waveforms of the 3-phase voltages.

S.C.

12

Figure 4.3: waveforms of the phase to phase voltages.

Let teta= Θ=wt

Where w = 2 x π x f [rad/s]

f: frequency of the AC supply.

The three phase voltages are given by:

Va= E x sin(Θ) eq 4.1

Vb= E x sin(Θ+120°) eq 4.2

Vc= E x sin(Θ-120°) eq 4.3

Phase to phase voltages are given by:

Vab = -Vba = Va-Vb eq 4.4

Vac = -Vca = Va-Vc eq 4.5

Vbc = -Vcb = Vb-Vc eq 4.6

In the following figure:

- P1, P2 and P3 represent the pulses on Thyristors T1, T2 and T3 respectively.

- Vmean: the mean voltage across the load

- α: firing delay angle

- VL: Voltage across the load as shown in figure 4.1.

S.C.

13

Figure 4.4: waveforms with a firing delay α.

Range of Thyristor Pair in conduction

+ 30o to + 90o T1 and T6

+ 90o to + 150o T1 and T2

+ 150o to + 210o T2 and T3

+ 210o to + 270o T3 and T4

+ 270o to + 330o T4 and T5

+ 330o to + 360o and + 0o to + 30o T5 and T6

The average voltage across the load is given by:

𝑉𝑚𝑒𝑎𝑛 =1

𝜋 𝑉𝐿𝑚𝑎𝑥 × sin 𝜃 𝑑𝜃

2𝜋

3+𝛼

𝜋

3+𝛼

=3

𝜋𝑉𝐿𝑚𝑎𝑥 cos(𝛼) eq 4.7

𝑉𝐿𝑚𝑎𝑥 = 3 × 𝑉𝑝ℎ

S.C.

14

4.2 Schematic of a 3-phase thyristor bridge on Matlab-Simulink

Figure 4.5: 3-phase Thyristor bridge on Matlab-Simulink

Same as for the DC motor model, the Thyristor bridge will be simplified by a subsystem.

Figure 4.6: 3-phase Thyristor bridge subsystem on Matlab-Simulink.

In figure 4.5:

- R, S and T (French indications) are the 3 phase voltage input supply also can be

referred to as R, Y and B (red, yellow, blue) or Va, Vb and Vc.

- V-, V+ is the output voltage of the bridge. (V- is to be connected to neutral/earth).

S.C.

15

4.3 Command of the 3-phase Thyristor bridge on Matlab-Simulink

The control of the 3-phase Thyristor bridge on Matlab-Simulink is done using a special block

called synchronized 6-pulse generator.

The synchronized 6-pulse generator takes as inputs the phase to phase voltages and the firing

delay α. The output is a vector that contains the six pulse signals.

In figure 4.7, the purpose of wiring is to create a 6-pulse generator subsystem that will take

the 3-phase line voltages and the firing delay α as inputs and generating the 6 pulses

separately as outputs.

The subsystem is displayed in figure 4.8.

Figure 4.7: Pulse generation with a firing delay α

Notes:

- Input “block” allows blocking the operation of the generator. The pulses are disabled

when the applied signal is greater than zero

- Frequency and pulse width are two parameters of the pulse generator that are to be

adjusted by double clocking on the block and modifying the corresponding values.

- P1, P2, P3, P4, P5 and P6 are Matlab selector blocks used to select the specified pulse

from the vector of pulses.

S.C.

16

Figure 4.8: 6-Pulse generator subsystem with a firing delay α

4.5 Simulation of the command – Thyristor bridge To test the command – Thyristor bridge, a simple resistor is used as a load.

The schematic is shown in figure 4.9 where the 6-Pulse generator subsystem of figure 4.8 and

the 3-phase Thyristor bridge subsystem of figure 4.6 are being used.

The simulation result at a firing delay α=0 is shown in figures 4.10 and 4.11.

The verification of the simulation is done by comparing the simulation results to figure 4.4

and the Thyristor bridge study in paragraph 4.1.

S.C.

17

Figure 4.9: testing of the command and Thyristor bridge.

S.C.

18

Figure 4.10: The 6 pulses triggering the Thyristor bridge (firing delay α=0).

Figure 4.11: waveforms (firing delay α=0) of the phase to phase voltages (Vab, Vbc, Vca)

and the output voltage of the Thyristor bridge (Vload at the resistance in this case).

S.C.

19

To simplify the system, in the following a “Triggered Thyristor Bridge” subsystem will be

used in the following. This subsystem (figure 4.12) combines the 6-Pulse generator

subsystem of figure 4.8 and the 3-phase Thyristor bridge subsystem of figure 4.6.

Figure 4.9: “Triggered Thyristor Bridge” subsystem.

This subsystem takes the 3-phase voltages (Va, Vb, Vc) and the firing delay α as inputs and

gives V+, V- as the output voltage of the Thyristor bridge.

The pulses are combined in one Vector output for display use only.

V- Simulation of the complete system: DC motor and inverter.

Figure 5.1: subsystem of the 3-phase supply source.

Explanation of figure 5.2:

1- 3-phase source supplies the Triggered Thyristor Bridge subsystem.

2- The firing delay α is set to 54 (random value).

3- The output voltage of the Thyristor Bridge is led to a LR circuit. The impedance-

resistance circuit acts as a filter to reduce the ripples in the output voltage.

4- After filtering, this voltage is used to supply the armature of the DC motor which is

modeled by a subsystem (as in figure 3.4).

5- The outputs to be studied are the current, torque and angular velocity of the DC

motor.

The simulation results are shown in figure 5.3 and figure 5.4.

By comparing these results to figure 3.5, figure 3.6 and figure 3.7, we notice that the curves

are the same except for the ripples in the curves of figure 5.3.

This phenomenon is caused by rippled output of the Thyristor bridge. Filtering has reduced

the ripples but not to total elimination. In paragraph 3, the DC motor was directly fed from a

DC supply. The comparison of input waveforms is shown figure 5.5

S.C.

20

Figure 5.2: Schematic of the DC motor and inverter model.

Va: armature voltage of DC motor Ia: armature current

Ce: torque Cr: resistant Torque

w or N: angular velocity

S.C.

21

Figure 5.3: closed-loop response - load operation, Torque and current in function of time.

Figure 5.4: closed-loop response - load operation, angular velocity in function of time.

S.C.

22

Figure 5.5: comparison of the input armature voltage between simulations in paragraph 3 and

paragraph 5.

VI- Control System

6.1 Cascade controller

In order to control the current and angular velocity, therefore two loops, a cascade controlled

will fit the application.

Figure 6.1: cascade controller

The equivalent Matlab schematic is shown in figure 6.2.

The first loop to be studied is the current loop followed by the angular velocity loop.

Before starting the study, design objectives need to be set. The controllers’ parameters are

studied to meet these design objectives.

S.C.

23

Figure 6.2: Matlab schematic - controlling system by cascade controller method

S.C.

24

6.2 Design objectives

For the current loop:

Response time tm = 0.35ms

Steady-state error εs < 4%

Percentage overshoot (PO) D% < 10%

For the angular velocity loop:

Response time tm = 0.2s

Steady-state error εs = 0

Percentage overshoot (PO) D% < 15%

Important Note:

Inverters of type chopper or Thyristor bridge are usually used with DC motors. Such type of

inverters is usually treated as a first approach to a constant gain called instantaneous

average when modeling the system. For further explanation, the output voltage is assimilated

to a mean value over a period of operation of the inverter.

This type of modeling is satisfactory only if the frequency chopping frequency of the inverter

is a lot greater than the frequency of the supply source. Otherwise, a different modeling

method should be applied where the sampling operation of the inverter is taken into

consideration and if necessary delays can be brought into the control.

6.3 Loop tuning steps

For loop tuning or gain parameters calculations, the procedure is as following:

1- Introducing a proportional gain to the Loop C1(p) K

2- Tuning K to manipulate response time, stability, Steady-state error and percentage

overshoot.

3- Introducing an Integral gain C p =1+τp

τp to decrease significantly the steady-state

error.

4- Tuning de 1/τ to a value << c where c represent the cut-off frequency of the open

loop of the system and control combined.

5- Introducing a Derivative gain C2 p = 1 + τp if the system stability was highly

degraded by previous gains.

Notes :

i- It is possible to take 1/τ < c /10, but in this case the settling time will increase

significantly and the response will be similar to the curve shown in figure 6.3.

ii- The effects of increasing a parameter independently is shown in the following

table:

S.C.

25

Parameter Rise time Overshoot Settling time Steady-state

error

Stability

proportional gain Decrease Increase Small change Decrease Degrade

Integral gain Decrease Increase Increase Decrease Degrade

Derivative gain Minor

decrease

Minor

decrease

Minor

decrease

No or slight

effect

Improve if

τ is small

Figure 6.3: Response curve featuring a large settling time.

6.4 Study of the current Loop

Figure 6.4: Block diagram of the current loop represented on Matlab.

Consider C(p)=I(p)/U(p) the transfer function of this loop.

C p =I(p)

U(p) =

1Ra + Lap

1 + 1

Ra + Lap × 1

f + Jp × Kmax2=

f + Jp

(Ra + Lap) × f + Jp + Kmax2

=1 +

fJ p

La. Jp2 + Ra. J + La. f p + Ra. f + Kmax2

S.C.

26

To simplify more, C(p) is written under this form:

C p =I(p)

U(p) = K0.

1 +p

w1

1 +p

w4 . 1 +

pw3

eq 6.1

Where

W1 =1

τm=

f

J= 0.5 rd/s

W2 =W3 + W4

2= Kmax2 +

Raf

LaJ = 44 rd/s

W3 =1

τe=

Ra

La= 37 rd/s

W4 =1

τem=

Kmax2

RJ= 52.5 rd/s

K0 =f

Kmax2= 7.56e− 3

U: armature voltage

I: armature current

Bode diagram asymptotes of the transfer function C(p) will take the following form:

Figure 6.5: Bode diagram asymptotes of the current loop transfer function.

S.C.

27

To get the bode diagram of the transfer function on Matlab, the following command will give

as a result the diagrams in figure 6.6:

Transfer function:

0.01512 s + 0.00756

----------------------------------

0.000515 s^2 + 0.046 s + 1

>> bode(H)

Figure 6.6: Bode diagram of the current loop transfer function

C(p)=I(p)/U(p) is the transfer function of current in function of voltage, to control the

current, a current controller is introduced to the loop. The current controller takes feedback

from output current and compares it to an input current also called the control variable that

the output current must follow.

As shown in figure 6.7, the current controller generates a voltage output that is led to the

Thyristor bridge model which was assimilated to gain Go as mentioned previously. The

output of this model is the armature voltage U.

Parameters of the controller are calculated by following the procedure of paragraph 6.3.

-50

-40

-30

-20

-10

0

Magnitu

de (

dB

)

10-2

10-1

100

101

102

103

104

-90

-45

0

45

90

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

S.C.

28

Figure 6.7: The block diagram of the current loop after applying a current controller.

The closed loop transfer function is given by:

H p =I(p)

Ic(p)=

1

1 +w1p

Kbo. (w2)2

eq 6.2

Where

Kbo = Kp × Go × Ko

Kbo: open-loop gain

Go: Thyristor bridge gain

Kp: proportional gain (controller gain)

Ko: C(p) gain

The closed loop transfer function H(p) is a first order function of the form:

H p =I(p)

Ic(p)=

1

1 + τp

With a time constant:

τ =w1

Kbo. (w2)2

To get a rising time tm= 0.35ms the cutoff frequency must be Wc = π/ tm = 8.8Krd/s.

This cutoff frequency is achieved when Kp.Go = 300.

The open loop gain is then Kbo = 2.268 (= 7.11dB).

S.C.

29

Figure 6.8: Bode diagram of the closed loop transfer function of current

To eliminate the steady-state error, an integral gain is added to the controller:

Gc p = 1 +1

Tic. p

Tic is calculated in a way of making the system close to order 1.

Let Tic = 1/W3=0.027.

In this case, the slop between W1and W3 (figure 6.5) becomes 0.

The simulation in figure 6.10 verify the calculations where tm = 0.35s and the steady-state

error is 0%.

Note: without PI, the steady-state error is 2% which is very acceptable. Therefore a

proportional controller can be used without the need to add and integral gain.

10-2

10-1

100

101

102

103

104

-90

-45

0

45

90

Phase (

deg)

0

10

20

30

40M

agnitu

de (

dB

)

System: untitled1

Frequency (rad/sec): 8.8e+003

Magnitude (dB): 0.00421

Bode Diagram

Frequency (rad/sec)

Cutoff

frequency

Wc

Kbo =

7.11dB

S.C.

30

Figure 6.9: Bode diagram of the closed loop transfer function of current after a PI controller

is introduced.

Figure 6.10: closed-loop response to a step input, current in function of time.

S.C.

31

6.5 Study of the angular velocity Loop

6.5.1 Block diagram, closed loop transfer function and applying a proportional gain

Figure 6.11: Block diagram of the angular velocity closed loop.

- First, the speed controller is considered to be a proportional gain Kp.

- The closed loop transfer function of current is calculated in paragraph 6.4 and given

by equation 6.2:

H p =I(p)

Ic(p)=

1

1 +w1p

Kbo. (w2)2 =

1

1 + τ′e p

- The mechanical model is given by 1

f+Jp

By replacing the blocks of figure 6.11 by these values, the result is shown in the following

figure:

Figure 6.12: Block diagram of the angular velocity closed loop with proportional gain Kp.

Open loop transfer function: Ω(p)

Ωc(p)=

1

f + fτe′ + J p + τe

′ Jp2

eq 6.3

Calculation of the closed loop transfer function:

HΩ p =Ω(p)

Ωc(p)=

Kbo′1

1 + τ′ep .

1 1 + τm p

1 + Kbo′1

1 + τ′e p .

1 1 + τm p

S.C.

32

HΩ p =Kbo′

1 + Kbo′ + τ′e + τm p + τ′ e . τm p2= k.

1

1 + 2. z.p

w0+

p2

w02

eq 6.4

Where

Kbo’=Kp/f: open-loop gain

k =Kbo′

1 + Kbo′

w0 = 1 + Kbo′

τ′e . τm

2. z =τ′e + τm

1 + Kbo′

1 + Kbo′

τ′ e . τm → z =

τ′ e + τm

2 1 + Kbo′ τ′e . τm

It seems to be a dilemma between stability and precision as an increase of Kbo’ in the

purpose of reducing the steady-state error leads into decreasing z therefore degrading system

stability.

Consequently, it is necessary to introduce a PI controller that eliminates the steady-state error

without highly affecting system stability.

Numerical application: K = 0.7, τ′ e = 3.69e-4

6.5.2 Applying a proportional - Integral (PI) controller

Returning to the open-loop transfer function of angular velocity given by equation 6.3:

Ω(p)

Ωc(p)=

1

f + fτe′ + J p + τe

′ Jp2=

1

0.01 + 0.02𝑝 + (7.378𝑒 − 006)𝑝2

Bode diagram of this transfer function is given in figure 6.13.

The rise time is calculated from the transfer function or can be deduced graphically by taking

the cut-off frequency Wc from the bode diagram given in figure 6.13.

Rise time = π/Wc = 0.05s

pour réduire le temps de monté qui est 0.05s a 0.02s par exemple on peut ajoute

un correcteur proportionnel qu’on peut encore calculer graphiquement :

S.C.

33

Figure 6.13: Bode diagram of the open-loop transfer function of angular velocity

The PI speed controller is given by:

C p = Kp(1 +1

τvp) = Kp

1 + τvp

τvp

To reduce the rise time from 0.05s to 0.02s for example, a proportional gain Kp is added in

order to make 157rd/s the new cut-off frequency.

Kp is calculated graphically:

At 157 rd/s the magnitude is -9.95 dB, thus 9.95dB must be added to make 157rd/s the new

cut-off frequency.

9.95 = 20 Log Kp ↔ Kp = 109.95

20 =3.14

As for τv, to simplify the system it can be given by:

1

τv=

Wc

10= 15.7 rd/s ↔ τv = 0.06

Thus the PI controller:

C p = Kp(1 +1

τvp) = 3.14(1 +

1

0.06p)

eq 6.5

As shown in figure 6.14, 157rd/s is now the new cut-off frequency of the closed loop transfer

function.

10-2

10-1

100

101

102

103

104

105

-180

-135

-90

-45

0

Phase (

deg)

-100

-50

0

50M

agnitu

de (

dB

)

System: G

Frequency (rad/sec): 49.9

Magnitude (dB): 0.0125

Bode Diagram

Frequency (rad/sec)

S.C.

34

Figure 6.14: Bode diagram of the closed-loop transfer function of angular velocity

6.5.2 Simulation results

Figure 6.15: closed-loop response to an angular velocity step.

10-2

10-1

100

101

102

103

104

105

-180

-135

-90

Phase (

deg)

-100

-50

0

50

100

150

System: untitled1

Frequency (rad/sec): 136

Magnitude (dB): 1.27

System: untitled1

Frequency (rad/sec): 157

Magnitude (dB): 0.0586

Magnitu

de (

dB

)Bode Diagram

Frequency (rad/sec)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

Angular

velocity

(rad/s) Ωc

Ω

S.C.

35

By analyzing figure 6.15, three parameters can be deduced:

Rising time = tm = 0.02s

Steady-state error = 𝜀𝑠 = 0

Percentage overshoot = D% = 6% (acceptable)

Figure 6.16: Current at no-load

I*: current at the output of the speed regulator, this current is the input of the current loop

where the current regulator must make the output current I of the current loop follow I*.

I: armature current (the output of the current loop)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.5

0

0.5

1

1.5

2

2.5

3

I*

I

Time (s)

Current

(A)

S.C.

36

Figure 6.17: Current and angular velocity response after adding a load (torque) of 4 N.m

starting at 0.5s.

The behavior is logical: after adding load to the motor’s speed (N) will decrease then the

regulator will increase this speed back to input speed of the regulator (N*).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Temps en s

Coura

nt

en A

I

I*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Temps en s

N e

n r

d/s

N

N*

Time (s)

Time (s)

S.C.

37

At load operation the motor will absorb more current since the armature current is directly

proportional to the torque (equation 2.2).

Annotations:

N=Ω

N*=Ωc : input speed or the speed at which the motor is required to run

I=Ia

I*=Ic (as shown in figure 6.7)

In the following a saturation block is added to the current. The purpose of adding a saturation

block is to limit the current Ic between 0 and 13A.

If the input speed is Nc = 100 rd/s and at 2.5s a load of 4N.m is added then the simulation

results will be as follows:

Figure 6.18: Current and angular velocity response after introducing a saturation block.

0 1 2 3 4 50

50

100

150

200

Vitesse e

n r

d/s

0 1 2 3 4 5-5

0

5

10

15

Temps en s

coura

nt

en A

N

Nc

Ic

I

Time (s)

S.C.

38

6.5.2 Summary of final results

Current:

- Rise time : 0.35s

- Steady-state error: 2% if proportional regulator and 0% if PI regulator.

- Percentage overshoot: D% = 0% (see figure 6.10)

- Current limitation by a saturation block.

Speed:

- Rise time :

0.02s without the current saturation block.

Between 0.02s and 0.25s with the current saturation block.

- Steady-state error: 0%

- Percentage overshoot: D% = 6% (see figure 6.15)

The final Matlab schematic of the system is shown in figure 6.19:

- 3ph source block: gives as an output the 3-phases voltages R,Y and B.

- These voltages are led to the Triggered Thyristor bridge block.

- The output Voltage of the bridge is filtered and used as the armature voltage of the

DC motor.

- The Load of the motor is represented by a step at input torque Cr.

- The output speed N is taken as a feedback and compared to a reference speed

N*=Nref=100 rad/s

- I* is the output of the PI speed regulator. I* is limited by the saturation block.

- The armature current Ia is then compared to I*.

- The output of the PI current regulator gives the reference voltage ec (as in figure 6.7).

- This reference voltage is translated to the firing delay angle α using equation 4.7:

𝑒𝑐 = 𝑉𝑚𝑒𝑎𝑛 =3

𝜋𝑉𝐿𝑚𝑎𝑥 cos 𝛼 =

3 3

𝜋𝑉𝑝ℎ cos 𝛼 =

→ 𝛼 = arccos 𝜋

3 3 𝑉𝑝ℎ𝑒𝑐

= arccos 𝜋

3 3 × 260𝑒𝑐 = arccos (2.24𝑒 − 3) × 𝑒𝑐

S.C.

39

Figure 6.19: Matlab schematic of the controlled system - a detailed version of figure 6.2.

I*

S.C.

40

Appendix

Numerical application of the project: 3-phase source:

Phase Voltage: V=260V.

Line Voltage: U=450V

Motor characteristics:

Nominal armature voltage: Un=260V

Nominal angular velocity: Nn=2150 tr/min= 225 rad/s

Armature inductance: La=34 mH

Armature resistance: Ra=1.26 Ω

Nominal Torque: Cn=14 N.m

Armature nominal current: Ian=13.5 A

Friction : f = 0.01

Inertia : J = 0.02 kg.m2

Matlab: MATLAB

® is a high-level technical computing language and interactive environment for

algorithm development, data visualization, data analysis, and numeric computation.

Matlab 6.5 is used for simulations in this document.

List of used subsystems:

1- DC motor model:

2- Thyristor bridge:

S.C.

41

3- 6-Pulse generator with a firing delay α:

4- Triggered Thyristor Bridge = Thyristor bridge + 6-Pulse generator with a firing delay

α:

5- 3-phase supply source: