Download 2.7.1 armature voltage control

Introduction:
DC Machine basics like d c. Motor, d c Generator, modes of
operation of DC Motors and DC Generators are discussed in
this chapter. DC Machines are part of Electrical Engineering.
If you are using any kind of stereo, Walkman, DVD player or
VCR you must observed a very small rotating motor inside the
appliance or instrument you are using. This is no doubt a DC
Motor. There are lots of application which can be completed
only with DC Machines, Traction is one among these.
1.1
Modes of Operation:
DC Machines are not outdated and basically a DC Machine can work
in three modes:
1.2
•
•
•
Generator Mode
Motor Mode
Break Mode
Fig 1.1 Basic DC Electrical Motor construction
1.2.1
Generator Mode:
A generator is a device that is used to produce electrical
energy or electricity. The main function of a generator is to
alter the form of energy. It converts mechanical energy into
electrical energy.
18
In dc. generators, the machine is driven by a prime mover with the mechanical
power and produce electrical energy. A dc. generator produce dc current as
output by using commutator and brushes.
1.2.2 Motor Mode:
A motor is just opposite in working and principle as compare to DC Generator.
A motor or dc Motor converts electrical energy [dc Voltage] into mechanical
energy or force.
1.2.3 Break Mode:
Break mode is something different from the above said mode. In break mode,
the machine ( which functions as a motor before the application of braking
action ) works as a generator and the electrical power developed is either
pumped back to the supply as in regenerative braking or is dissipated in the
machine system as in dynamic braking. Hence in the braking mode the machine
deadlerat on account of power supplied or dissipated by it and, therefore,
produce a mechanical braking action.
1.3 Classification of DC Machine Based Upon Power Intake:
Based intake power type DC Machine is of two types viz. DC Motor and DC
Generator.
Type
Power Input
Power Output
DC Generator
Mechanical power
Electrical power
DC Motor
Electrical power
Mechanical power
18
2.1 Introduction
DC motors have been used in industrial applications for years. Coupled with
a DC drive, DC motors provide very precise control. DC motors can be used
with conveyors, elevators, extruders, marine applications, material handling,
paper, plastics, rubber, steel, and textile applications to name a few.
2.2 Construction
Fig.2.1 DC Motor construction
DC motors are made up of several major components which include the
following:
• Frame
• Shaft
• Bearings
• Main Field Windings (Stator)
• Armature (Rotor)
• Commutator
• Brush Assembly
18
• For servo applications DC drives is still popular because of good dynamic
response and ease of control.
2.7 Speed control DC motor methods:
The speed control of a separately - excited DC motor is investigated in details.
The steady state equivalent circuit of separately-excited DC motor is shown in
Fig. 2.3. The armature circuit voltage equation and mechanical behavior
of separately -excited DC motor at steady state are:
(2.22)
Va = Ra la + E
E
=ce<j>n
(2.23)
Tei = Cm d> Ia
(2.24)
It is more convenient to normalize the above equations with suitable
base value to obtain dimensionless equation. The rated armature voltage (Van)
is taken as the base value of the voltage, while the short circuit current at rated
Voltage (Isc) is taken as the base value of the current. The corresponding short
circuit torque is the value of the torque.
(2.25)
The base field flux is the rated field flux (<J)n) corresponding to rated filed
current (Ifm) at rated field voltage (Vfm). The base motor speed is chosen to
be the no load motor speed (no). Neglecting the armature voltage drop at
load, the no load speed is obtained from
(2.26)
Vam — Ce (j) r no
The motor normalized speed and current are obtained to
n/n0 - Va/Van (cp / cpn
f
(2.27)
- Ra/Ran (<p / <Pn )" (Te/Tsc) la/ Isc
2
(2.28)
= ( 9 / <Pn )"’ (Tel/Tsc)
18
The motor speed given by eq. (2.27) depends on; the armature voltage (Va), the
armature circuit resistance (Ra) including both armature and additional series
resistance circuit and the motor field excitation ($).
In practice, for speed less than the base speed, the armature current and field
currents are maintained constant to meet the torque demand, and the armature
voltage (Va) is varied to control the speed. For speed higher than the base
speed, the armature voltage is maintained at the rated value and the field current
is varied to control the speed. Figure.2.8 shows the characteristics of torque,
power, armature current, and field current against the speed.
The speed control of a DC motor is achieved by varying one of these variables
[1]. The speed control methods are discussed in the following item.
2.7.1 armature voltage control
The main field flux maintained constant at its rated value, the armature voltage is
varied in the range [ -1.0 < (Va/Van)< 1.0]. When rated armature voltage is applied
and the field current is fixed to its max. value , the motor speed is called the base
speed (nt,).
The armature applied voltage is controlled by using rotating machine converter as
in ward-leonard system or static converters or a combination of them. Static
converter for the armature voltage control is achieved either by a controlled
rectifier or by a DC chopper control.
The controlled rectifier is fed form an AC source, and the voltage is changed by
adjusting the value of the firing angle. The controlled rectifier is associated with
AC voltage harmonics and additional power loss. The DC chopper is fed on the
contrary from a DC source, which may be the output of an uncontrolled rectified
fed from the AC supply.
The value of the output voltage is controlled by adjusting the chopper circuit onoff ratio or the chopping frequency. The DC chopper output voltage is associated
with lower output voltage harmonics when compared with the controlled rectifier.
Armature voltage control gives the ability to load the motor by its allowable
maximum torque without overloading of the motor. The motor normalized speed
and current are obtained to the speed-torque and current-torque characteristics of
the separately -excited DC motor with armature voltage control are shown in
Fig.2.9 a and b respectively.
a- Speed-torque characteristics
b- Current-torque characteristics
Fig. 2.9 : Speed-torque (a) and current-torque (b) characteristics with armature voltage control.
21
2.7.2 Field weakening control
The field weakening control is used when it is required to obtain motor speeds above
the base speed or when the armature voltage cannot be changed. The armature voltage is
fixed at the rated value and the field current and consequently the main flux is reduced.
Maximum speed obtained by field weakening is limited by the mechanical design of the
motor and the motor stability at weak field flux due to the magnetizing effect of
armature reaction.
The motor normalized speed and current are obtained to the speed-torque and currenttorque characteristics of the separately-excited DC motor with field weakening control
are shown in fig.2.10 a and b respectively.
Fig. 2.10 : Speed-torque (a) and current-torque (b) characteristics
with fieid weakening control.
2.7.3 Armature resistance control
A variable additional resistance is inserted in the armature circuit. This method is
accompanied by additional copper losses especially when the motor speed is greatly
decreased resulting in very poor motor efficiency. So, this method is economically
relevant for small motors and in drives with narrow speed control ranges operating for
short periods in the reduced speed range. The motor normalized speed and current are
obtained to speed-torque characteristics of the separately- excited DC motor with
armature resistance control is shown in Fig. 3.5.
Fig. 2.11 : Speed-torque characteristics with armature resistance control.
3.1 Introduction
In many industrial applications, it is required to convert a fixed voltage DC
source into a variable voltage dc source of controllable average value. A dc
chopper converts directly from dc to dc and is also known as a dc converter. Dc
chopper are used usually where the source is a battery as in industrial
electrically powered trucks and where it is a third rail or over heat trolley wire
as in rapid transit cars. They provide smooth acceleration control, high
efficiency and fast dynamic responses.
Chopper are used to replace switched, series armature circuit resistors, the
advantage is higher efficiency continuous control, and the ability to operate the
motor in regenerative braking mode and it can be used in dc voltage regulators
and also used in conjunction with an inductor to generate a DC current source,
especially for the current source inverter.
When a dc drive is used, then there are the following possibilities:
a- The use of current converter circuit with low reactive power
consumption, for example, half-controlled bridge,
b- Tapped-transformer with adjustable secondary voltage and an
uncontrolled rectifier.
3.2 Principle of operation of DC chopper
Fig.3.1 shows the equivalent circuit and the corresponding output voltage of a
DC chopper. In this circuit the chopper is represented by on-off switch. When
the switch CH is closed for a certain time (ton), the input voltage appears
across the load. When this switch is opened for another certain time (toff), the
voltage across the load drops to zero. During a specified time period T, the onoff ratio of the chopper is controlled, this results in controlling the average
output voltage across the load.
Fig. 3.1: chopper circuit configuration and output voltage
wave form The average output voltage is given by:
V a v = ^j;° n V s d t = t - f V s = K V s = t o n FV s
= (l-toffF)Vs
(3.1)
The average voltage can be adjusted either by controlling the duty cycle of
the chopper while keeping the frequency constant (pulse width modulation)
or controlling the chopper frequency (pulse frequency modulation) while
keeping certain (on-off) ratio.
3.3 methods of control of output chopper
voltage 3.3.1 Pulse width modulation
In this technique the chopping frequency f is kept constant and the on-time and
the off- time of the chopping switch is varied as shown in Fig.3.2 (a), this
method has many advantages such as,
1.
2.
Easier isolation.
Signal filtering due to constant frequency operation.
(a) Pulse width modulation
(b)
pulse frequency modulation
Fig. 3.2 :(a) principles of PWM & (b) pulse frequency modulation (PFM)
3.3.2 Pulse Frequency Modulation (PFM)
The on-time or the off-time of the chopper is kept constant and the frequency is
varied as shown in Fig.3.2 (b). To obtain a full output voltage range using the
frequency modulation control, the frequency is varied over a wide range,
consequently, harmonics at new frequencies would be generated and the filter
design became difficult.
3.4 Classification of DC chopper circuit
DC chopper circuits are classified according to the power flow between the
source and the load. The direction of the power flow depends on the forced
path for the power and the type of the load.
3.4.1 First-Quadrant chopper (Class A chopper)
The circuit arrangement of the first-quadrant chopper is shown in Fig.3.3(a).
The current flows from the source to the load and the chopper circuit operates
as a rectifier. This configuration is used for motoring operation of a DC motor.
(a)Chopper circuit configuration
(b) Voltage-current characteristics
Fig. 3.3: Configuration (a) and voltage-current
characteristics (b) of first-quadrant chopper
Both load current and voltage are positive, i.e. operation is in the first quadrant
as shown in Fig.3.3 (b). During the on-period the chopper CHi is switched on
for a time interval ton. During the off-period the switch CHi is switched off for
a time interval Wf, the armature current will flow in the closed loop via the
freewheeling diode Di.
3.4.2 Second-Quadrant chopper (Class B chopper)
The circuit arrangement of the first-quadrant chopper is shown in Fig.3.4(a).
The current flows from the machine to the source. The machine voltage is
positive, i.e. operation is in the second quadrant. The chopper circuit operates
as an inverter and is used for regenerative braking of DC motor.
(a)Chopper circuit configuration
(b) Voltage-current characteristics
Fig. 3.4: Configuration (a) and voltage-current characteristics (b) of secondquadrant chopper
During the on-period the chopper CH2 is switched on for a time interval ton ,
and the load voltage vanishes Vo=0.0. During the off-period in which the
chopper CH2 is switched off for a time interval t0ff, the energy stored in the
magnetic circuit is returned to the supply through the diode D2.
3.4.3 Two-Quadrant Type-A chopper (Class C chopper)
The circuit arrangement of this type is shown in Fig.3.5(a). the load voltage
is positive and the load current is either positive or negative.
(a) Chopper circuit configuration
(b) Voltage-current characteristics
Fig. 3.5: Configuration (a) and voltage-current characteristics (b) of Two-quadrant
type-A chopper
The load current state depends on the switching case of the choppers CHi and
CH2. When chopper CHi is operated and diode Di is used, the circuit operates
in the first quadrant (motoring). When chopper CH2 is operated and diode Do is
used, the second quadrant operation is obtained (braking). It must be avoided to
switch both CHi and CH2 simultaneously to prevent short circuit on the source.
The voltage-current characteristics are shown in Fig3.5 (b).
3.4.4 Two-Quadrant Type-B chopper (Class D chopper)
The circuit arrangement of this type is shown in Fig.3.6 (a). The load current is
positive and the machine voltage is either positive or negative. This
configuration allows to operate either as a rectifier as an inverter.
(a)Chopper circuit configuration
(b) Voltage-current characteristics
Fig. 3.6: Configuration (a) and voltage-current characteristics (b) of Two-quadrant
type-B chopper
When the choppers CHi and CH2 are turned on, the power flows from the DC
source to machine in the shown current direction (motoring). But when
chopper CHi and CH2 are switched off, the load current will continue to flow in
that path created by the diodes D1 and D2. The power flows from the machine
to the DC source in serves direction to the current path shown (regeneration).
3.4.5 Four-Quadrant chopper (Class E chopper)
The circuit arrangement of this type is shown in Fig.3.7 (a). It is the general case
in which the load current or the machine voltage may be positive or negative
according to the switching choppers. The voltage - current characteristic are
shown in Fig.3.7 (b) which shows different modes of operation
(a) Chopper circuit configuration
(b) Voltage-current characteristics
Fig. 3.7: Configuration (a) and voltage-current characteristics (b) of Four-quadrant
chopper.
By controlling the sequence of switching of the choppers (CHi, CH2, CH3&CH4),
one can obtain the desired case of operation, i.e. operation is possible with
reversible regenerative. This chopper is the basis for the single phase bridge
inverter and it is characterized by circuit complexity rather than its high cost.
.4 Types of DC Motors
The dynamic behavior of the DC machine is mainly determined by the type of the connection
between the excitation winding and the armature winding.
2.4.1 separately-excited DC motor
The equivalent circuit of separately-excited DC motor is shown in Fig. 2.3
O------------------->—
+ iI
a
!
;P ii
12
B
Fig. 23: equivalent circuit of separately-excited Dc motor
When a separately excited dc motor is excited by a field current and armature current flows in
the armature circuit, the motor develops a back electromotive force (emf) and a torque to
balance the load torque at a particular speed. The field current of a separately excited dc motor
is independent of the armature current and any change in the armature current has no effect on
the field current. The field current is normally much less than armature current. The equations
describing the characteristics of separately excited dc motor can be determined from figure 2.3.
12

(2.1)
(2.2)
£
v f = Rf if + L f (% )
05
II
The instantaneous equations of the separately excited DC motor are
Va = R a ia + La ( % ) + e
(2.3)
T'd ~if i- a
(2.4)
T a = J {% ) + Bo + T L
(2.5)
• Under steady state condition, the time derivative in the above equations are zero
and hence, the steady-state average quantities are:
(2.6)
(2.7)
E = Kv a) If
Va —
I a Ra Eg
=
Ra ^I f
Td = f y if ia = + T L
(2.8)
(2.9)
The developed power is
(2.10
)
The speed of separately excited DC motor can be found from
V a -R a Ia
60 —
KvIf
(2.11
)
2.4.2 Self excited DC motor 2.4.2.1 Series DC motors
In the Series motors, the field windings are connected in series with the armature circuit. Figure 2.4
presents the equivalent circuit of the series DC motor. The field circuit is designed to carry the
armature current.
Fig. 2.4: The equivalent circuit of the series DC motor
The speed of the DC series motor is determined and can be varied by controlling the
armature voltage or armature current meanwhile, the demand torque is determined as
the following.
^a—
I a
( Ra~/o
K-v *a
Td= KfIfla=B(o+ TL
0) =
—
— (2.12)
(2.13)
The series motor can provide a high torque especially at starting and for this reason
DC series motors are commonly used in traction applications. For speed up to the
base speed the armature voltage is varied and the torque is maintained constant.
2A.2.2 Shunt DC motors
The field circuit is independent of the armature circuit because both circuits are fed from voltage
source. Armature current and speed depend on the mechanical load.
Fig. 2.5: The equivalent circuit of the shunt DC motor
2A.2.3 Compound DC motors
Compound- wound motor which has two field winding, one connected in parallel with the armature and the
other in series with it. There are two types of compound motor connections. When the shunt field winding is
directly connected across the armature terminals [Fig. 2.6 a], it is called short-shunt connection. When the shunt
winding is so connected that it shunts the series combination of armature and series field [Fig. 2.6 b], it is called
long-shunt connection.
2.5 Speed /Torque characteristics
The following chart compares speed/torque characteristics of DC motors.
At the point of equilibrium, the torque produced by the motor is equal to the amount of torque required to turn the
load at a constant speed. At lower speeds, such as might happen when load is added, motor torque is higher than
load torque and the motor will accelerate back to the point of equilibrium. At speeds above the point of equilibrium,
such as might happen when load is removed, the motor’s driving torque is less than required load torque and the
motor will decelerate back to the point of equilibrium.
The DC drive system is designed to control and to vary the speed and torque of the DC motors and they span wide
range from a very simple designed hard ware to a very complicated programmable software microcontroller kits.
DC motor drives have many advantages such as:

The DC drive is relatively simple and cheap compared to induction motor drives nut DC motor itself is

more expensive.
For low power applications the cost of DC motor plus drives is still economical.
2.3 Theory of operation
When voltage is applied to the motor, current begins to flow through the field coil from the negative terminal to
the positive terminal. This sets up a strong magnetic field in the field winding. Current also begins to flow
through the brushes into a commutator segment and then through an armature coil. The current continues to flow
through the coil back to the brush that is attached to other end of the coil and returns to the DC power source.
The current flowing in the armature coil sets up a strong magnetic field in the armature.
Fig. 2.2: (a) Armature and commutator segments, (b) Armature prior to the coil's wire being installed, (c) Coil of wire
prior to being pressed into the armature, (d) A coil pressed into the armature. The end of each coil is attached to a
commutator segment.
3.5 separately excited DC motor fed from DC chopper
To study the performance of the chopper circuit, its operation is divided in to two main modes. During the on
mode, the chopper is switched on and the current flows from the supply to the load.
35
During the off mode, the chopper is switched off and the load current continues to flow through freewheeling
diode (D).
•o
£>f
Chopper
L
A
D
7WD
Fig.3.8: DC chopper fed separately excited DC motor
3.5.1 Mode of operation
3.5.1.1 Continuous armature current
During the on mode (1) (0< t < ton) the voltage equation is then given by
Ks= i?aia+ L a ^ + E
(3.2)
And its solution gives the load current
(3.3)
_
(3.4)
At t= 0 — I (0) = I
a
so
35
iai® = *±(l-e~tba)+lsoe~t'Ta
At the end of model (t=ton) the current attains the initial value for mode2
iDo = iai(t) = ^7(1 - e_t°n/T“) + Iso e During mode 2, ton < t < T or Tm
The load is short circuited through the freewheeling diode and the voltage
equation is then given by:
35
At the end of mode2, the chopper is turned on again for the next cycle
after the time period,
T = ton + toff
(3-8)
The current at the end of mode2, is the initial current for the next cycle
h = ia(ton + toff) = IDo e
/t«J (3.9)
With the duty cycle
(3.10)
la
3.5.1.2 Discontinuous armature current
u=0 (3.n)
To know if the motor operates in the discontinuous armature current we
calculate at first "tm" if
a: tm < T we have discontinuous armature current b: tm > T we have
continuous armature current At, ton < t < T or Tm
35
FI
ton)/ \ (P
ton)(
ia(t)=j-a[ 1-e /t*) + IDO e ^
(3.12)
We assume the motor operate in the discontinuous armature
current At t— tm , ia(tm) =0
0 = -t(l~e
/Ta) + lD0 e ^
trn = Ta ln[l + —j£-
(3.13)
E(e ,Ta-l
3.5.2 Motor performance 3.5.2.1 Average Armature Voltage
Raia+ La^ + E
(3-14)
Vs dt = Raia+ Ladia + Edt
(3.15)
By integration from (0 to t) and dividing by T
1 rton
R, rT
L
flDO
r^DO ^ fton
[T
- Vsdt = — I iadt + -^{ iadt + I dia] + — { I Edt + Edt]
1 Jo
‘a Jo
la
Jiso
Jiso
I a J0
Jton
35
Vst-f=Raia + 0 + Et-f
(3.16)
Vst-f = Raia + Et-f+E-E
(3.17)
Vst-f = Va + Et-f-E
(3.18)
Vs = VsD + E(l-tf)
(3.19)
35
3.5.2.2
Average Armature current
Ia=Ya fo Wdt
3.5.2.3
Torque
T= Cra . Ia for continuous flux
3.5.2.4
Torque / speed characteristic
The motor used to obtain the speed - Torque characteristic is
Ce=0.17, Vs = 200v, Ra = 3ohm, n=1000rpm, La= 0.06H , Fch=
500HZ
The flow chart of the program used to calculate this characteristic as
in figure 3.10
T
Fig. 3.10: speed - Torque characteristic
35
3.5.3 Ripple Armature current
Under steady- state condition, Is0=l3 and the peak -to-peak load ripple
current can be determined to
ton
(7~~ton)
A/a = ho ~ ho = T1—T
Ra
—
(3'23>
1 -e~a
The maximum peak-to-peak ripple current at duty ratio D=0.5 and is
given by
A Ia max = ^ ■ Tanh-|Ka
(3-24)
4 Ia
For the switching period t« 4 Ta this expression for the maximum
excursion reduce to Vs T
AIa max = TT ■ 77^ (3-25)
a max Rg 4Tg
The armature current excursion at steady operation condition can however
be approximately obtained from the voltage -time area applied to the
inductance La of the armature circuit. The average output voltage is given
by
Ys. = £°2l = D
(3.26)
vs T
And the voltage across the motor inductance is given by
35
VL(t)=Va(t)-Va(0)
(3.27)
V L(t)=V a(t)-V a(to„) (3.28)
A /« = ^ = £ (T - t„„) = f ^ U(3.29)
^a *a
The voltage-time area applied the inductance La is given by
VL.T0n=La. A I a
(3.30)
From which the current excursion is obtained to
35
For a duty ratio D=0.5, the same maximum current excursion is obtained as in above equation . This equation can
be used to determine the inductance of the armature circuit necessary to limit the current excursion to a preset
value usually a value of 10% of the armature current tolerated.
It is also obvious that the switching chopper frequency fCh should be as high as possible to reduce the load
current excursion and to reduce the size of any additional series inductors in the load circuit the chopper is also
limited.
4.1
DC Chopper operation
4.1.1
First-Quadrant chopper (Class A chopper)
4.1.1.1 Chopper output controlled by Pulse frequency modulation.
Fig. 4.1 First-Quadrant chopper (Class A chopper) controlled by 555 timer.
Fig.4.1 shows the first-Quadrant chopper (Class A chopper) controlled by 555 timer circuit. The output
of 555 timer is the firing of the S19CF power transistor. Fig.4.2 is presents the firing pulse train with
different frequency which can be controlled by varying the CT from 0.1 |xF to 0.01 |iF.
Fig. 4.2 Pulse frequency modulation (PFM)
Fig.4.3 shows the output voltage of the chopper which will be applied on the motor armature. It can be
seen that the armature voltage changed by operating the chopper by PFM which tested in Fig.4.2.
35
Fig.4.3 The output voltage of the chopper(Armature voltage) with respect to the
firing pulse train(PFM).
The yellow pulse is related to the chopper output voltage (armature voltage) and the green pulses is
related to the chopper firing signal.
35
4.1.1.2 Chopper output controlled by Pulse width modulation.
Fig.4.4 is presents the firing pulse train with different width which can be controlled by varying the
RA in the circuit shown in figure 3.1 from 1 kQ to 2 kQ.
Fig.4.4 pulse width modulation(PWM)
35
Fig.4.5 shows the output voltage of the chopper which will be applied on the motor armature. It can be
seen that the armature voltage changed by operating the chopper by PWM which tested in Fig.4.4.
Fig.4.5 The output voltage of the chopper(Armature voltage) with respect to the
firing pulse train(PMM).
The yellow pulse is related to the chopper output voltage (armature
voltage) and the green pulses is related to the chopper firing signal.
4.1.2 Second-Quadrant chopper (Class B chopper)
The circuit arrangement of the second-quadrant chopper is shown in
4.6 Second-Quadrant chopper (Class B chopper)
Fig.4.6. The current flows from the machine to the source. The
machine voltage is positive,
i.e. operation is in the second quadrant.
35
Fig.4.7 The output voltage of the chopper(Armature voltage) with respect
to the firing pulse train in Second-Quadrant chopper (Class B chopper)
The yellow pulse is related
to the chopper output voltage
35
(armature voltage) and the green pulses is related to the
chopper firing signal. The chopper circuit operates as an inverter and
is used for regenerative braking of DC motor.
4.1.3 Comparison between (Class A chopper) and (Class B chopper)
ig.4.8 clarify the in vers operation relation between Class A and Class B
Fig.4.8 Class A and Class B output chopper(armature voltage)
4.2 Determination of Speed -torque characteristics
Fig.4.9 shows the linear Speed-torque curves of separately excited DC motor fed by DC Chopper with
various value of duty cycle D at continuous current mode.
D _ Va _ T-Ton
(4 j)
Vs T
Va= D Vs
(4.2)
Vn—E
From Eq. 3.21 (Chapter-3) average current: Ia = ——
a
35
At duty cycle D the Eq. will be: la = ~~R~ (43)
Ka
From Eq. 3.22 (Chapter-3) Torque:
Td= K . Ia for continuous flux (4.4)
From Eq. 4.3 and 4.4
Td = k(Dvs-E2 (4 5)
Ra
E = kwd
(4.6)
From Eq. 4.5 and 4.6
Td = (4,7)
wd = £r-%r<i
=|
(4.9)
Tan = ^
35
From Eq. (4.8),(4.9) and (4.10):
(4.11)
(4.12)
From Eq. 4.9 and 4 . 1 1 :
Wd
= Dw dn-—wdn
1
dn
d — £) _ 1A.
dn
^dn
n
n
(4.13)
Where: Wd=27aid ,and Wdn=27tndn We will use the Eq. 4.13 to
plot the
speed torque characteristics at various value of duty cycle D (0.2, 0.4, 0.6, 0.8 &1):
MATLAB CODE:
%Wd=2*Pi*nd;
%Wdn=2*Pi*ndn;
%n=nd/ndn=Wd/Wdn;
%T=Td/Tdn;
%Wd/Wdn=Duty cycle -Td/Tdn;
%The analysis for continuous current mode;
%The duty cycle is 0.2,0.4,0.6,0.8,1; clc
clear all n=(0:0.01:1) ;
T=l-n;
n_l=(0:0.01:0.8);
T_1=0 . 8-nJ; n_2=(0:0.01:0.6) ;
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T_2=0.6-n_2; n_3=(0:0.01:0.4) ;
T_3=0.4-n_3; n_4=(0:0.01:0.2) ;
T_4=0.2-n_4; figure (1)
plot (n, T, ' b' , n_l, T_1, 'b',n_2,T_2, 'b \n_3,T_3, 'b',n_4,T_4, 'b')
xlabel('Td/Tdn') ylabel ( ' nd/’ndn ' ) axis([0 1.2 0 1.2])
..--No load speed.
Max. Power
-
\ x X* _ 0.6
-
0.8
|
0
T3
N. Nv ' -------------------------- \ _
* \ v.
1 I. \ \ \
■
\ \ 1\ '■
\ 111
C
0.4
\1
0.2
0
0.2
1.2
0.4
0.6
0.8
1
Fig.4.9 Speed-torque curves of separately excited DC motor fed by DC Chopper with various value of
duty cycle D at continuous current mode
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11