small-signal anal yzing of sep aratel y excited dc motor

sAÜ Fen
Bilimleri Enstitüsü Dergisi
3. Ci lt 2.Sayı
( 1999) 77-82
SMALL-SIGNAL ANAL YZING OF SEPARATELY EXCITED DC MOTOR
FED DC DC CHOPPER
-
Isınail COŞKUN
•
Gazi Üniversitesi, Teknik Eğitim Fakültesi Elektrik Eğitimi Bölümü, Ankara-Türkiye
ABSTRACT
I. INTRODUCTI O N
Choppers are widely used in order to get
smooth speed characteristics of DC motors. If the
load or input voltage is changed, a feedback
control technique is used to get the constant speed.
To design the proper feedback control, the transfer
function of the chopper-motor combination should
be kno\vn.
ln this study, the effort
was
given to obtain
the general black diagram, inciurling the field
current and aı1nature current of chopper-fed DC
motor. It can be seen from the black diagram that
the chopper circuit modifies the transfer function
of the DC motor.
DC-DC DARBELE YİCİ
YABANCI
iLE
UYARTIMLI
KÜÇÜK SİNYA L ANALİZİ
BESLENEN
DC
MOTORUN
,
tasarlamak
widely
used
for
speed
offer
high efficiency, quick response, wide speed
control range and regeneration down to very low
speeds [ 1] .
All chopper circuits can be classified
into two groups. (a) Load independent choppers�
in w h ich the output voltage waveform is either a
square wave or can be approximated by a square
\vave. (b)
Load dependent
choppers, in which
charging of the commutating capacitor is govemed
by load current. In such cases, the output voltage
wavefoım is neither a square wave nor can be
approximated by a square wave. Various methods
of analyzing of DC motors fed by a chopper with
square
wave
output
voltage
are
reported
references [2-4].
1
sho'\\7S
a schematic circuit
Uygun bir geribesleme
iç�
darbeliyici-motor
chopper.
+
Chopper
Gain K
kombinasyo nunun transfer fonksiyonu bilinmelidir.
-
Bu çalışmada, darbeleyici ile beslenen DC
motorun
alan
akımı ve endüvi
akımını
da
kapsayan genel blok diyağraınını elde etıııek için
çaba
gösterilmiştir.
görülebileceği
gibi,
Blok
darbeleyici
diyagramından
devresi
DC
motorun transfer fonksiyonunu değiştirmektedir.
Fig. 1 Chopper fed separately
excited DC motor
77
--
in
diagram of a DC separately excited motor fed by
DC motorlardan düzgün bir hız karakteristiği
etmek
için
elde
darbeleyiciler çok
kullamlır.
giriş
gerilimi
Eğer yük ya da
değişirse sabit hız elde etmek için geribeslemeli
kontrolu
are
control o f D C separately excited motors as they
Figure
ÖZET
kontrol tekniği kuJlanılır.
Choppers
Smaii-Signal Analyzing of Separately Excited OC Motor Fed DC-DC Chopper
KE
In this DC motor back emf coefficient
,
is considered constant and does not change
\vİth the annature current.
The perfoıınance
equations of the motor can be \vrİtten as belovv
[5].
Vdc K
d
R.
ala + La dt
Tem
-
Where,
Vde
=
ea
-
K
=
,.r
Tm
em
TL
=
=
-
F
-
(1)
.
+ eg
ıa
(2)
KE if COm
(3)
Kr if İa
d
(4)
J dt
ID
m
+F
O> m
+ TL
(5 )
K !!Vdc
d
�Tm= J dt Aro
m
+F�rom+ô.TL
(10)
If we take Laplace transfoıın of these
equations, where the Laplace variabtes represent
only the small-signal � values in equation 6
through 10.
Lllia(s)
(ll)
RaMa(s) + sLaMa(s) +
=
=
KE(lrodrom(s) +
ffimo(s) Mt(s))
�Tem(s) =
(6)
(8)
(9)
Iao �İr)
Kr(IroM (s) + laoMf(s))
a
(12)
(13)
(14)
These equations for the motor-load
represented
can be
combination
by
transfer-function blocks as in Figure 2. As it can
be seen in Figure 2, inputs to the motor-load
combinations are the field c urrent
Lllı{s).
annature teınıinal voltage �Vdc(s) and the load
torque �TL(s).
Fig. 2 Open-loop block diagram of the chopper fed
separately-excited DC motor.
78
&r)
�Tem = KT �İr Ll İa = KT(Iro � ia +
Lllig(s)
-
=
mo
�g(s)
,
a
= K E(IrcArom+ro
Lllia(s)
supply voltage
armature voltage
duty-cycle
electromechanical torque
mechaırical torque
load torque
viscous friction constant.
For analyzing
small-signal dynamic
perfoıırıance of the motor-load combination around
a steady-state operation point the following
equations can be written in terms of smail
de"Viations around their steady state values [6].
�e
�eg = KE &r 1\rorn
used.
II.THE C H OPPER-MOTOR
The chopper-motor combination is shown
Figure 3. It is a step-down chopper, having
utstanding
feature
ability
and
to
start
onunutating reliably. In this chopper, Tı i s the
1ain switch and T 2 is the auxiliary switch. L1 and
-= are closely coupled inductors.
. The input vector
. The output vector
. The state variabtes veeter
Flux proportional to ir.
time
is
very
short
comp ared to the switching period
so
can
it
X
Chopper
is
operating
continuous
mode.
When the
main s\vitch
series inductance is
\\rhen
the
==
main switch
inductance is La
L1
L
=
Where,
Cx
y
as
in
the
Tı is ON, the
L1 + La and
is OFF, the
values.
Under these conditions, the block d iagram
' the chopper-fed separately excited
DC motor
ill be de veloped
.
T
u = (�Vde
.ôTL)
T
y = (�ea , ı:lTem)
x = (&a, &f, Llrom)T
+
X
+ Du
dt
matnxes.
(16)
Bu
dx
•
•
L.
La have close
and
Ax
•
T,
neglected.
be
are
The state-space equations are,
Assumptions:
Commutating
vectors
below.
COMBINA TION
1
The considered
'
(17)
A, B� C and D are
During the operation period, there are two
modes of operation. When switch Tı is closed, a
positive Ioop occurs, mode 1 a n d
\Vhen
switch T1 is
open, zero loop occurs, mode 2 . Therefore, two
sets of equation will be required.
Mo de 1: Positive loop.
When T1 is closed, the equivalent circuit
is become in Figure 4.
c
Tı
Fig. 4 The equivalent circuit when
Tı is closed
The first mode of operatio� the following
Fig.
equations
3. Chopper-motor combination
l. THE BL O C KDlAGRAM OF C HOPPER
FEDDC MOTOR
Ystern�
In order to develop the block diagram of
the state-space averaging method will be
are
obtained.
(18)
(19)
(20)
(21)
79
Smaii-Signal Analyzing of Separately Excited OC Motor Fed DC-DC C hopper
(22)
follo'ving
the
B, C and D, ınatrixes,
18 through 22 are rearranged in
o
state equation foın1.
-R
-
ıleg
=
- KEromo
a
Lt+
La
KTiao
J
A==
ı
ı
o
matrixes,
ol
o o
Aı,
tl1e equations
rearranged in state
'
ı
o
B,
o
-
23
through
equations
D2
=
Tı is open
If the state-space averaging model is used, the
B
=
I
and
27
a
foıın.
o
-1
J
'
'
Fig. 5 The equivalent circuit when
K Aı + (I-K) Aı
K Bı + (1-K) Bı
(24
(25
Cı
Bı,
i
-
�eg
ı
A
Mode 2: Zero loop.
When T 1 is ope� the equivalent circuit
is b ecome in Figure 5.
A
+
KE(Iro�IDm + �comofur)
To obtain the
'
dt
Ai a
'
J
c1 =
Ra&a + La
d
(23
(2c
o
-ı
obtained.
are
o
-
thı
operation�
of
mode
equations
�ea
To obtain the A,
the equation
second
The
A, B,
C
and
(28)
(29)
C
D
o o
o o
D
==
=
matrixes are obtain ed
K Cı +(1-K) Cı
KD1+(1-K)Dı
as
follO\\rs�
(3Cı
(31
The state-space averaged matrixes a re b ecome as follows.
80
ll
. Coşkun
,
-
R
•
A=
1 -K
K
+ L
(L
)
L
+
1
a
a
F
J
--
B
D
K
Lı + La
-
o
o
-
K
o
-
ı
c
'
J
-
-
o
o
KTifo KTiao
o
o
'
O
o
The averaged matrixes can be in the state-space equations .
X
•
y
==
-
Ax
Cx
+
+
Bu
Du
(32)
1
L\eg
a
La
dt
+I- K]Lle
g
La
Lı +La
--
(33)
(34)
=
(35)
(36)
Transfoıntıing equations
Lllia(s)
=
UVc1c(s)
32
through
36 in the Laplace transform gives;
(37)
(39)
(40)
81
Small-Signal Analyzing of Separately Excited OC Motor Fed OC-OC Chopper
�Tm(s)
==
sJ �ro m
(s)+ F�rom(s)
+
�TL(s)
(41)
Equation 37 through 41 can be put in block diagram in figure 6.
transfer function changes.
�Vde(s)
_.
---.
... K
AEa(s)
• _J_"\
From fig.6 it can be seen that the
ı
� �,....__
+ .4�ı
..
....__-----1 KE Ifo ""
Acom(S)
Fig. 6 The open-loop block diagraın of separately
excited DC motor fed DC-DC chopper
IV. RESULTS
The block diagram for the chopper fed DC -DC
separately excited DC motor was obtained using
the state-space averagin g method.
By using the
block diagram in Fig. 6, the transfer function
be obtained and analyzed.
When the transfe
function is obtained, the chopper circuit effect
should
be
taken into account.
can
V. REFERENCES
[1]
K.
dynamic
B. Naik, et al.," Steady state and
response analysis of a
chopper controlled
DC
separately excited motor'' IEEE Trans. on Power
Apparatus and Systems, Vol. PAS-104, No.7, pp.
1775-1782,july 1985.
[2]
K.
B.
chopper
Electric
Naik,
fed
DC
et al., "Dynaınic
separately
Machines
and
model o f
excited
motor"
Electromechanics,
[4]M.H.RashicL
"A
Thyristor
Internationaljournal of Electronics� Vol. 53, No.l
pp.71-81, 1982.
(5] J. Best and P. Mutschler," Control of arınatur
and field current of a chopper fed
drive
rd
by a
IFAC
single
chip
Electrical
Switzerland, pp. 515-522, 1983.
state
[6]
82
Feb., 1971.
motc
SymposYum on Control in Po,ve
[3] R. Pari melagam and V.Rajagopalan, "Steady
Appl., Vol. IGA-7, pp. 101-108,
DC
ınicrocomp uter�'
Electronics
with separate excitation" IEEE Trans.Ind. Gen.
\\if
minimum limits on voltage control of DC drives
pp.497-5 16, Dec.,l 982.
i nvestigation of a chopper fed DC motors
chopper
and
Drives,
Lausannc
N. Mohaı\ T. M. Undeland and at al, Po,Ye
Electronics, John Wiley and Sons, Ine. 1989: p·
286-2 9 5.
'•