Prom. Nr. 3442
Investigation
of
a
thyratron-
motor
by
or
thyristor-controlled synchronous
simulation
analog computer
on an
THESIS
presented
to
THE SWISS FEDERAL INSTITUTE OF TECHNOLOGY, ZURICH
for the
degree
of
Doctor of Technical Sciences
by
Mohamed Mahmoud
B.Sc. Electrical
Bayoumi
Engineering
Citizen of Egypt
Accepted
on
the recommendation of
Prof. Ed. Gerecke and Prof. A. Dutoit
Buchdruckerei Berichthaus
Zurich 1963
Leer
-
Vide
-
Empty
Investigation
of
a
thyratron-
motor
or
thyristor-controlled synchronous
by simulation
on an
analog computer
Leer
-
Vide
-
Empty
To my Parents and to my wife
Leer
-
Vide
-
Empty
Acknowledgment
I
am
greatly
his valuable
abled
me
to
My thanks
consent
indebted to Professor Ed. Gerecke for
suggestions
accomplish
are
and
helpful guidance
that
en¬
this research.
also due to Professor A. Dutoit for his
to examine this thesis.
Contents
Notations
10
1. Introduction
16
Preliminary experimentation
2.
18
2.1
Machines and instruments used
2.2
Single-phase investigation
2.2.1
2.3
Connection
18
diagram
18
2.2.2
Oscillograms
21
2.2.3
External characteristics
23
Three-phase investigation
2.3.1 Connection diagram
2.3.2 Oscillograms
3. Mathematical
4. Determination
24
24
27
analysis of the synchronous machine
The machine
3.1
18
equations expressed in
of the machine
4.1
Stator resistance and
4.2
Field time constant
4.3
Direct and
4.4
Determination of the
the
per-unit
30
form
...
36
constants
leakage
36
reactance
36
quadrature axis synchronous
leakage
reactances
37
Direct axis transient and subtransient reactances and time
4.5
39
constants
Determination of the leakage coefficient and time constant
4.6
in the
quadrature
5. Machine
6.1
8
36
coefficients in the direct axis
direction
6.
34
as a
axis direction
42
synchronous generator
Analog Computation of
the
single-phase
47
controlled
synchronous
machine
53
Mathematical formulation
53
6.2
Simulation of the inverter effect
59
6.3
Analog set-up
Analog Computer
60
6.4
results
60
6.4.1
Oscillograms
60
6.4.2
External characteristics
63
6.4.3
Current locus
66
Analog computation of the three-phase controlled synchronous
7.
machine
69
7.1
The mathematical formulation
69
7.2
Machine with variable
82
7.3
The
speed
thyratron simulator
7.3.1
Block
83
diagram
87
7.3.2
The "saw-tooth" generators
87
7.3.3
The
89
7.3.4
The
coinciding voltages
Schmitt trigger
7.3.5
The
pulses determining
7.3.6
The bistable multivibrator
90
7.3.7
The
90
7.3.8
89
the
end-points
of the currents
gate
Advantages of the thyratron simulator
94
Analog investigation
8.
9. Simulation
of the system
90
95
on
the PACE
9.1
The mathematical formulation
9.2
Oscillograms
9.3
Characteristic
analog computer
....
100
100
112
curves
124
9.3.1
External characteristics
124
9.3.2
Current locus
124
9.4
Behaviour of the air gap flux vector
124
9.5
Transient behaviour
128
10.
Summary and conclusion
139
Zusammenfassung
141
Literature
143
9
Notations
'i> h> h
if
—
=
flowing
spectively (in ampere)
in
tne stator currents
the stator current
pressed
in the
by the
re¬
phase
in
No. 1 when it is
ex¬
No.
per-unit system
AH the currents,
terized
flowing
1, 2 and 3
phases
voltages,
letter
r
fluxes and resistances are charac¬
when they
are
expressed
in their rela¬
tive form
if
iD
flowing
=
the current
=
the current
in the field
flowing
in the
winding
damper winding in
the direct
axis direction
iQ
=
the current
flowing in the damper winding in
the qua¬
drature axis direction
ia
=
the current
the
a
flowing through
replace the
direction to
hypothetical winding
the
stator. The
a
in
direction coin¬
cides with the direction of the phase No. 1, i.e. it is fixed
with respect to the stator
ip
=
the current
flowing through
hypothetical winding
the
in
direction that is also fixed with respect to the stator
the p
windings.
The
/? direction
advances that of phase No. 1
by
90°
i0
=
the
id
=
the current
the
the
stator. The d direction coin¬
zero
sequence current
flowing through
d direction to replace the
hypothetical winding
in
cides with the axis of the rotor
iq
=
the current
flowing through the hypothetical winding
the q direction which advances the d direction
/„
-yjl
id-c.
amplitude
=
the
=
the current
smoothing
h.c.
Ils
—
=
10
of the stator current at normal load
flowing
from the d.c.
inductance
(refer
to
voltage Ud through
idc.
the fundamental component of the stator current
with sin y
the
Fig. 1)
the average value of the current
phase
in
by 90°
it in
Ilc
=
the fundamental component of the stator current
phase
Ud
=
a
d.c.
with
cos
voltage
y
chronous machine
through thyratrons
"i» u2, u3
the
phase voltages
«,',
=
the
voltages between
synchronous
power supply
the
u'0
=
the
voltage
point
u'w
=
Uf
=
used to feed the stator of the syn¬
source
=
«2, M3
of the
synchronous
the three terminals of the stator of
d.c.
voltage
synchronous
point with respect
used to feed the field
source
amplitude
negative
winding
of the
phase voltage
ut in
phase voltage
ut in
phase voltage
the
Uu
=
the fundamental component of the
phase
to the middle
machine
=
=
of the d.c.
with respect to the
U„ yJ2
Ulc
point
battery
the voltage of the star point
pole of the battery
a
machine in volts
machine and the middle
of the star
of the
iL in
of the normal
with sin y
the fundamental component of the
phase with
cos
y
winding
in ohms
R
=
resistance of the stator
Rf
=
resistance of the field
RD
=
resistance of the damper winding in the direct axis direction
—
resistance of the
RQ
winding
damper winding
in the
quadrature
axis
direction
Tf
=
the field time constant
TD
=
the
damper winding time
TQ
=
the
damper winding
=
the direct axis
synchronous inductance
=
the direct axis
synchronous
reactance in ohms
xd
=
the direct axis
synchronous
reactance in per unit
Lqq
=
the
quadrature
axis
synchronous inductance
Xq
=
the
quadrature
axis
synchronous
reactance in ohms
=
the
quadrature
axis
synchronous
reactance in per unit
Lfd
=
the mutual inductance between the field coil and the d coil
LDd
=
the mutual inductance between the D coil and the d coil
LDf
=
the mutual inductance between the D coil and the field coil
Lid
l
Xd
xq
constant
(coil D)
time constant
(coil Q)
11
LDD
Lff
Lqq
LQq
V-&
=
the self-inductance of the D coil
=
the self-inductance of the field
=
the self-inductance of the
=
the mutual inductance between the
Q
winding
coil
Q coil and the
q coil
LdDLfi
=1
—
Ltd LfD
LfpLdf
Vf
Liir-Li,
'AD
uD
LDf LdD
=1
T
J--DD
7«f
_
LifLfd
j
LffLdd
—
JdD
1
^dD ^Dd
_
Lod Ldd
LDf LfD
.
JDf
<7„
*
x"d
I
Wf
_
_
LqQLQq
=11
IT
=
the direct axis subtransient
synchronous
reactance in per
unit
synchronous
reactance in per unit
x'd
=
the direct axis transient
Xt
=
the
leakage
=
the
quadrature axis subtransient synchronous
jc^'
reactance in ohms
reactance in
per unit
U'q
=
the
quadrature axis subtransient synchronous inductance
ftu ,/'2> ftz
=
the flux
linking
the stator coils 1, 2 and 3
tjff
=
the flux
linking
the field
"Adj "Ag
=
the ^ux
unking
the D and
^a> ftp
=
tne
^UX
linking
the
coils
respectively
the flux
linking
the d and q coils
respectively
<A</> ftq
}j/0
—
=
the
zero
and
Q coils respectively
/?
sequence flux
t
=
time in seconds
t
=
the
12
a
per-unit
respectively
winding
value of the time
a'
y
frequency of the synchronous machine
=
the normal
=
the
=
3
=
the
firing angle measured
the
phase No.
synchronous speed
U„I„
angle
the
between the axis of
1 and the rotor axis at the
the current
flowing
=
conduction
angle
=
the
angle
as
in
phase
No. 1 (see
between the axis of the
point of begin of
Fig. 19)
phase No.
1 and the rotor
axis
»*
V
=
=
the instantaneous value of the
=
the
M.
angular synchronous speed
pairs
=
the number of
=
friction coefficient
=
angular speed
of
"
>
poles
moment of inertia
=
mechanical time constant
=
mechanical torque
=
electromagnetic torque
o
M
M
=
=
L'ej
M-
R
0
0
0
0
0
0
0
-xq -1
0
10
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
R
0
*„
1
1
0
0
0
0
0
0
1
0
0
0
0
0
'xd
i
[L']
1
TD-xd(l-adD) TD
w
Tf-x,(l-<rif) 7>(1-~Hf)
0
1
0
0
TD(l-n„)
0
0
7}
0
0
0
0
0
0
xq
0
0
0
Ta xq(l -".) ?<
The reference
quantities
whole of this thesis. The
their
=
corresponding
taken by Laible [3] are used throughout the
following table gives the physical quantities and
reference values.
13
Physical
Reference value
value
U.y/2
voltages
(a)
stator
(b)
stator currents
W2
(c)
stator flux
,
•A„=——
Vmy/2
linkages
co„
(d)
stator
(e) active,
R-U"
impedances
reactive and total power
3
h
In
3
Psn
(f) torque
(g) speeds
ns
ton
angular speeds
(h)
time t
(i)
field current
(j)
field flux
=
t
=
CO.
2nt
Unj2
<*>n'Lfd
Umy/2m
linkage
o>»
(k)
current in the
damper winding in the
direct axis
(1)
flux
linking
the
current in the
quadrature
(n)
damper winding
iin the
14
damper winding
in the
axis
linking the damper winding
quadrature axis
flux
Lff
ht
Unj2
(On'hd
direct axis
(m)
[/„•/„
Unj2
LDD
<»n
hi
Unj2
(°„-LQq
in the
Unj2LQQ
ton
hi
Where the
per-unit system
is not
applied,
the
Giorgi system of
been used. The recommendations for mercury
national Electrotechnical Commission
arc
(IEC) [19]
and the list of
the Association of the Swiss Electrical
suggested by
have been adopted, especially:
[V]
volts
[mV]
[A]
[mA]
[VA]
[kVA]
[HP]
[s]
[ms]
[Hz]
[rev/min]
millivolt
units has
vapour of the Inter¬
symbols
Engineers (SEV) [18]
ampere
milliampere
volt-ampere
kilovolt-ampere
horsepower
seconds
milliseconds
Hertz
=
cycle/second
revolutions per minute
15
1. Introduction
Since the invention of the first electric motor,
control the
d.c. motor
of these
speed
of- that motor. It is
smoothly
and at
disadvantages
will, however it has
is that there is
one
possible
a
of the aims
to vary the
some
was
speed
disadvantages.
maximum limit
on
to
of the
One
its power and
applied voltage. The voltage is limited because it is mostly applied
on the rotor through brushes and segments. On the other hand, the syn¬
chronous motor has a fixed speed, namely the synchronous speed, while
the maximum limit put on its power and on the applied voltage is much
greater than those corresponding to the d.c. motor.
It is desirable to combine the advantages of the synchronous motor and
on
the
those of the d.c. motor. This
motor, being
a
a
synchronous
supply through inverters
d.c.
be achieved
can
by the
from
or
an a.c.
supply through
rectifiers and another of inverters.
Thyratrons
tifiers
rectifiers. The
easy,
be used
can
as
inverters
while the limit put
power is
high.
There
noncommutator
machine in construction and fed either from
on
or
a
set of
silicon-controlled
rec¬
control is
fairly
speed
applied voltage and on its
resulting from the commutation as
the maximum
troubles
are no
or
those involved with the d.c. motors.
Another advantage
from
a
d.c.
venient
supply
place
presented by
is that the
the noncommutator motor
thyratron
tubes
can
operating
be located at any
and when silicon-controlled rectifiers
are
used,
then
a
con¬
very
small volume is necessary.
The noncommutator motor offers
and
improved insulation
promise
a.c.
as an
over
adjustable speed
motors. In its
advantages
behaviour,
motor of
The
place of the
speed
simplified
construction
great flexibility against classical
the motor is
motor, where the group of thyratron tubes
takes the
of
the conventional d.c. motor. It shows great
analogous
to
a
commutator
(or silicon-controlled
rectifiers)
commutator.
variation of this type of motors
can
be executed
by changing:
(a)
firing angle of the inverter tubes,
the
(b)
magnitude of the field current,
(c) the amplitude of the applied d.c. voltage, which is effictively done by
changing the firing angle of the rectifier tubes when the motor is fed
the
from
16
an a.c.
supply,
or
by
(d) changing
the
triggering frequency of
the
source
controlling
the in¬
verter tubes.
The aim of this thesis is to
and
mathematically.
investigate this type of motors experimentally
The Park's
equations
of the
analysis. It is
windings, one in
is the basis of the mathematical
coils will be
replaced by
two
axis and the other in that of the
neglecting
assumed that the
machine
damper
the direction of the direct
quadrature
axis. The
assumption of
pole machine
the effect of saturation is also made. A salient
is considered and the variation of the permeance
is assumed to follow
stator
synchronous
windings
are
a
over
cosine function of the rotor
assumed to be
sinusoidally
the average value
position angle
y. The
distributed in space.
17
2.
Preliminary
experimentation
2.1 Machines and instruments data
DSM
three-phase synchronous
-
machine
serial number 383075 MFO
145/250 V, 8 A, 3.5 kVA
1500 rev/min, 50 Hz
The machine is provided with
-
-
either in star
or
4 salient
poles. The
stator can be connected
in delta.
GMn
-
-
-
125 V, 97
1400
rev/min
serial number 313966 MFO
Three-phase
-
-
-
A, 14 HP
tachometer
220/380 V, 0.5/0.29 A,
1500 rev/min, 50 Hz
190 VA
serial number 27673 Hiibner
Thyratrons
-
AEG,
2.2
ASG 5155
Single-phase investigation
2.2.1 Connection
Fig.
A; mercury vapour, 12.5 A
diagram
1 shows the connection
diagram
of a
synchronous machine,
one
of its
supply through thyratrons. Starting
performed with the help of a d.c. machine which is made to run at a
speed so high that, when the field winding of the synchronous machine is
phases being
fed from
a
d.c. power
is
fed,
One
a
back
phase
voltage
controller. Thus
nect the
18
may be available at its stator.
three-phase tachometer
it is possible to switch on
of the
driving
d.c. machine.
is used to feed the
the
voltage Ud
thyratron
and to discon¬
3
o
~
IU
»
nrV\_@—tzl
J^—[J4
120V OHz
V
o-
R5y
120 V OHz
-o
1
Fig.
1
diagram
Connection
of the
single-phase
controlled
-
2 3 45678
synchronous
machine
flowing in
1
=
stator
/'„
=
current
2
=
rotor
/j
=
phase
current
3
=
d.c. machine used for
=
angle
between the axis of the
R6...R8
=
shunts
4
=
d
=
direct axis
5
=
=
quadrature
variable resistors
switches
starting
or
loading the synchronous
machine
Ld
I...IV
ud
=
=
=
=
=
"i
=
=
three-phase
tachometer
thyratron controller
smoothing inductance
q
Rl...R5
=
thyratrons
Sl...S4.
=
d.c. voltage source for feeding the stator
voltage drop across the smoothing inductance
voltage across one of the thyratrons
9-10, 11-12, 13-14
phase voltage
d.c. voltage source
for
feeding the field winding of the
syn¬
chronous machine
=
y
current
.flowing
in the
smoothing inductance
19
=
one
of the
axis
connected to CRO
thyratrons
phase
No. 1 and the rotor axis
Leer
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Vide
-
Empty
Oscillograms
2.2.2
Fig. 2 shows oscillograms of the
current
if,
the
across one
voltage
of the
across
thyratrons
the
stator current
smoothing
iu phase voltage
uu field
inductance uL and the
«„. It is to be noted that the induced
voltage
voltage
in the stator
winding is no more a sine wave and that the distortion ap¬
only during the interval at which current flows in the stator winding.
Also during this interval, the field current i{ will be affected, but it is un¬
pears
affected in the interval
at
which
no
current flows in the stator
winding.
(a1) 100VJ—L
(a)
(a2)
6A
+—75 ms.—+
(b)
Fig.
Ud
Oscillograms
2
=
127 V
a'
(al)
=
phase voltage
(b)
=
thyratron
(cl)
(dl)
(e)
(f)
=
=
=
=
of the
=
single-phase controlled synchronous machine
140°
(a2)
uy
current
if
=
=
2.18 A
phase
current it
iel
d.c. current id-c>
phase voltage wx
(c2)
field current if
phase voltage ut
(d2)
voltage uvl across the thyratron
voltage uL across the smoothing inductance
=
=
21
(10) "AOOL
(?)
(2°)
V9
(IP) ~l_TAOOL
(P)(
(2P)
(a)
(J)
33
V9'l "L_T
"L-TAOS
AOQ
2.2.3 External characteristics
3 shows the external characteristics of the
single-phase machine at
speed (1500 rev/min). T^eapplied voltage Ud and the d.c. current
id-c- are plotted with the firing angle a' as parameter. A voltmeter and an
ammeter (Siemens) whose class of accuracy is ±0.2 and whose pointer
Fig.
normal
accuracy of 0.1 %
possesses
an
angle
measured
was
on a
were
used for the measurement. The
to increase the accuracy of measurement. The
angle
a' is
10
Fig.
nous
3
given
20
30
in the notations
40
firing
Tektronix CRO, using the 5X magnifier in order
50
60
70
on
definition of the
firing
page 13.
%
External characteristics of the
single-phase controlled synchro¬
machine
1
a'
=
170°
4
a'
=
140°
2
a'
=
160°
5
a'
=
130°
3
a'
=
150°
6 a'
=
120°
23
Three-phase investigation
2.3
2.3.1 Connection
diagram
Fig. 4 shows the connection diagram of a synchronous machine, with the
phases fed from a d.c. power supply through 6 thyratrons connected
in a bridge circuit. Starting is performed as explained in 2.2.1, where the
three-phase tachometer feeds the thyratron controllers. Since no two
conducting pulses come at the same time, a provision should be made
so that the current finds a closed path at the moment the switch St is
three
closed. This
allows the
to
provide
was
achieved
thyratron
another
by constructing
the circuit shown in
grid
controller which feeds the
pulse
at
the
same
time
as
the
pulse
of
Fig.
5. This
thyratron
No. 1 is
No. 2
produced.
R,
+
200V
o-
o3
T
I
Ri
I
o4
<±\
1o
y
U,
R„
1
Fig.
5
same
Circuit
time
as
producing pulses
the
pulses controlling thyratron
uG
=
the output of
uG.
=
connected to the
T
=
1/2
Ri
=
47 kO
1
=
from
2
=
from cathode of
3
=
to
4
=
to cathode of
24
thyratron controller
Ecc82
grid
grid
at the
of
grid-cathode
Rk
=1.5 kQ
R2
=
thyratron
of
No. 1
grid of thyratron No.
thyratron
2
No. 2
thyratron
1
No. 2 at the
No. 1
No. 1
thyratron
No. 2
Ra
47 kO
thyratron No.
of
=
39 kQ
9
10
o
FlF^i
+
u
n
120 V OHz
!i_ji_!f
V
o-
-4mA)—r-rh—o
+
120V OHz
1
Fig.
4
Connection
diagram
of the
three-phase
controlled
2 3
A
5 6 7 8
9 10 11 12
synchronous
machine
1
=
stator
4
=
current
2
=
rotor
it
=
phase
3
=
d.c. machine used for
=
angle between
=
shunts
starting
loading
or
the
synchronous
machine
R6...R8
4
=
5
=
Ld
=
smoothing inductance
I...VI
=
thyratrons
Ud
—
uL
=
uv
=
ul, u2, w3
=
Uj-
—
three-phase
thyratron
d.c.
tachometer
d
controller
voltage
source
for
feeding the
voltage
source
for
feeding
the field
winding
of the syn¬
chronous machine
'd.c.
=
current
flowing
in the
smoothing
inductance
25
one
of the
thyratrons
current
the axis of the
=
direct axis
quadrature
Rl...R5
=
variable resistors
Si... S4
=
switches
9-10, 11-12, 13-14
stator
in
=
q
voltage drop across the smoothing inductance
voltage across one of the thyratrons
phase voltages
d.c.
y
flowing
=
axis
connected to CRO
phase
No. 1 and the rotor axis
Leer
-
Vide
-
Empty
2.3.2
Oscillograms
Fig. 6 shows oscillograms of the
controlled
synchronous
currents and
voltages of the three-phase
machine. It is to be noted that the
voltage u'0 of
point with respect to the mid-point of the d.c. supply is a pulsating
voltage whose fundamental frequency is equal to three times that of the
phase voltage.
the star
(a)
(b)
(c)
(aU
50V
(a2)
6A
(bl)
50V
(b2)
6A
(d)
50V
(c2)
6A
l*-20msH
27
(dl)
50 V
(d)
(d2)
'
_
15A.
50V
(e)
k^Omd
(f)
28
20V_
(g)
f*-20msj|
Fig.
Oscillograms
6
of the different currents and
phase controlled synchronous
Ud
150 V
=
(al)
(bl)
=
=
(cl)
=
(dl)
=
(e)
(0
(g)
=
=
=
phase voltage
phase voltage
phase voltage
=
voltages
of the three-
machine
1.75 A
=
140°
ut
(a2)
=
phase
ux
(b2)
=
thyratron
current
i\
current
/Bl
the d.c. current idiC-
(c2)
«t
field current ix
(d2)
phase voltage ut
voltage across the thyratron uvl
voltage across the smoothing inductance
the voltage u'ok of the star point measured with respect
tive pole of the battery Ud
=
=
to the nega¬
29
3. Mathematical
The
analysis of the synchronous machine
general theory and mathematical description
machine
were
founded in 1929
by
Park and others
of the
synchronous
[1]. They established
theory used for handling rotating machines in general, where
voltages and fluxes are transformed along two axes. These
may be fixed in space or they may be rotating with the rotor.
the two-axis
all the currents,
two
axes
The choice of the first type of axes proves to be
to
cylindrical
axes
are
general,
advantageous when applied
rotor machines and to induction machines. The movable
adequate
the fixed
apply
to
axes are
and the movable
axes
machines with salient
on
used when there is
an
pole
rotors. In
unsymmetry in the
stator
when the unsymmetry lies in the rotor. In the
when both the rotor and the stator
are
symmetric,
either type of
case
axes
may be used.
synchronous
motor is
phases dis¬
placed around the stator at 120° interval, whereas the rotor comprises the
field winding and the damper coils, the axis of the field being in line with
that of the rotor, while the damper coil maybe resolved into two windings,
one in the direct axis direction and the other in the quadrature axis direc¬
The stator of the
composed
of three
tion.
windings (1, 2, 3) will be replaced by
The three stator
dings (a, P) which
to it
as
shown in
Fig.
fictitious win¬
fixed in space, the first
are
with the stator coil No. 1, the second
In addition to the a, /?
two
(the a coil) being in line
(the P coil) being perpendicular
7.
directions,
there is the
the rotor axis. Thus all the stator
quantities
zero
sequence direction
will be transformed
along
along the
p and zero sequence directions. The next step is to transform from the
p directions to the d, q directions. Hence it is possible to reduce the
machine into two sets of coils, one along the d direction and the other
along the q direction. The following equations are valid for transient or
steady state currents and voltages, and whether these currents and voltages
a,
a,
are
sinusoidal
or
not. The
30
=
of the armature
are:
^-
(1)
R- i2+-—
(2)
u1=R-i1 +
u2
voltage equations
at
at
Fig. 7
Transformation of a
stationary system
and to
three-phase stationary system into two-phase
two-phase rotating system
axes
fixed with respect to the stator
=
axes
moving
ut,u2,u3
=
stator
'u *2> '3
=
ux, Up
=
ud,uq
=
a,
fi
d,
q
=
if
iq
=
=
=
D
=
Q
with the rotor
phase voltages
stator phase currents
equivalent voltages in the fixed axes coordinates
equivalent voltages in the moving axes coordinates
equivalent currents in the fixed axes coordinates*
equivalent currents in the moving axes coordinates
current in the damper winding in the direct axis direction
current in the damper winding in the quadrature axis direc¬
tion
Uf
if
y
=
d.c.
=
field
=
angle between
voltage
source
used to
supply
the field
winding
current
the axis of the
phase No.
1 and the rotor
axis
«3
=
K-
i3 +
Those of the rotor
Uf
=
Rf-if
dty3
(3)
dt
windings
,
+
d^i
dt
are:
(4)
31
0
=RD-iD+-£
(5)
=RQ-'l<2+-7T
(6)
at
0
dt
2, 3 directions into the
The transformation from the 1,
takes
place according
/.
2
ip
=
j0=
*.=
to the
i2 +
h\
(7)
—f=02~h)
(8)
(/1 + /2 + f3)
-
l
(9)
[^i-^r^]
(10)
(11)
(^1+^2 + ^3)
^0=3
(12)
In order to transform the currents and fluxes from the a,
d,
q
directions, the following equations
id
=
ig
=
^
=
ilfq
=
32
are
/? directions into
used:
/.-cosy + ^-siny
(13)
-4-siny+i^-cosy
(14)
i/vcosy+i/vsiny
(15)
-t/ra-siny + ^-cosy
(16)
Now the flux
the
y? directions
following equations:
=4=^-^3)
V3
*,
the
a,
likages \jid, \\if, yjD
are
related to the currents
id, if, iD by
equations:
i>i
=
\j/f
=
<Ad
=
Am
'd + A/i
•
Z,^ j,,+J:#
•
Aid
•
*<i +LfD
•»/+Aw
•
•
*>+Ad/
if +ldd
'd
•
(17)
Id
(18)
'd
(19)
The
corresponding equations
in the
quadrature
are:
iJ/q=Lqq-iq+LQq-iQ
(20)
•Afl
(21)
=
l<lQ. h+LQQ lQ
•
The inverse transformation from the
tions will take
place according
d, q directions into the
a.,
ft direc¬
to:
(22)
=
id-cosy-iq-sin y
=
id
<A«
=
"Ad-cosy-i/^-siny
(24)
t//f
=
ij/f
(25)
4
if
sin y +
iq
sin y +
cos
xl/q-
(23)
y
cosy
/? directions
following relations:
In order to transform from the a,
tions,
axis direction
we
have the
='* + 'o
h
=-2««+^-'/» + 'o
h
original 1, 2, 3 direc¬
(26)
'i
1
to the
Ji
(27>
1
Jl
=-^L-^y h + h
(28)
<Ai=^ + ^o
1
(29)
Jl
^2=--^x+^Y^p + ^o
1
V3=
(30)
Jl
-^'-—^fi + ^o
(31)
We note that the transformation is made from the
original
axes
to the
rotating d, q
directions with the aid of the transformation along the fixed a,
P directions. This has the advantage of greatly reducing the necessary num¬
ber of multipliers necessary in the simulation
on
the
analog computer.
calculating the different currents may be done if all
machine voltages, currents, fluxes, resistances and inductances are put in
Another method for
a
matrix form. This method
problem
was
simulated
on
was
also
chapter
analog computer.
applied
the PACE
in
a
later
when the
33
3.1 The machine
equations expressed in the per-unit form
general practice, when analyzing synchronous machines to specify
quantities in relative form rather than to use the absolute
It is the
all machine
physical quantity is obtained by dividing
corresponding reference one. A list showing all
values. The relative value of any
its absolute value
by
the
reference values used in this thesis is to be found
ing equation (1) into
general
sponding
Applying
ones
(32)
changed, however, all values
this
on
the
0
=
0
The
=
=
expressed
we
in rela¬
get the
corre¬
equations (1)...(3), (7)...(16),
general appearance, but the equations (4),
dyf
i>+r;
(33)
d-z
?D+TrD
ra+ra
.
(34)
dx
dVQ
'
(35)
dz
equations (17) ...(21)
will be reduced to:
V*-
=
xi-ird + iD+irf
(36)
vf
=
*„•(!-•»</)-«;+(i -A*/) irD+irf
(37)
Vd-
=
*„•(!-ffiD)"'i+iji> + (1--/*!>)•'/
(38)
vq
=
Xq lq + Iq
(39)
VQ
=
V(1- V"'? + 'Q
(40)
'
where
rf
0>nL<M
xd
J?
=
Rf
n=
t:
Q
34
are
equations (1)...(31),
in their relative form. The
(22)...(31) will not change the
(5), (6) will be reduced to:
»/
page 10. Transform¬
dx
form is not
tive form.
on
yields:
?p-
u\=R'-ir1 +
Its
the relative form
<»»LDD
Rd
0}„LQQ
R
Q
Xq~
Rn
MnLqq
Rn
The
leakage coefficients
are:
LqQLQq
(41)
LqqLQQ
Ldf L'fd
°df
(42)
LffLdd
Lad L*Dd
C<iD
(43)
=
Ltd Ldd
L/d L/d
/V
'-•Df
Hd
LdD
(45)
Ldf LDD
Vd =1-
These values
(44)
LffLiD
LdD L/d
are
related to
Ltd Li
(!-*</)
(46)
l^dd LfD
=
one
another
by
the
following equations:
(47)
_
^=(1-^(1-^)
(48)
35
4.
Determination
of the machine constants
4.1 Stator resistance and
leakage
The stator resistance
measured
and it
The
=
0.0571
reactance of the stator
leakage
Potier
triangle
X,
xr,
method
1.03 Q
=
Rr
using the voltmeter-ammeter
corresponding to 75 °C.
transformed to the value
was
Thus R
was
reactance
method and
=
1.18 Q
=
0.0653
was
winding (x,)
was
measured
using
the
found to be,
4.2 Field time constant
The machine
excited
at
no
so
made to
was
that the stator
run at synchronous speed and the field winding
phase voltage is equal to half the rated voltage
load. The armature is left unloaded and the field
short circuited
termined from
Tt
T}
=
=
on
a
0.37
itself,
and the armature
voltage
of the armature
semi-log. plot
winding is suddenly
recorded.
s
quadrature
axis
synchronous
reactances
cording
to the American Institute of Electrical
A short
description of
slip
our case
voltage
36
was
speed
applied
adjusted
made
ac¬
test codes
[7].
is
the
was
given in
quadrature axis synchronous reactances,
test was made. The machine
was
Engineers
quantity
the measurement of each
As for the direct and
the
de¬
voltage.
The measurement of the reactances and time constants
the
was
116.18
4.3 Direct and
following.
Tf
was
to be 1470
driven at
rev/min.
a
very small
slip,
in
Subnormal sinusoidal
to the armature terminals with the field open. The
current
flowing in
the stator
value when the rotor axis
and
reaching
respect
to
that of that stator
=
armature
voltage
per-unit armature
current
—
quadrature
=
synchronous
axis
-
armature
with
4.4 Determination
T'D
=
0.5805
10.47 Q
voltage
=
0.2684
=
4.84 Q
of the leakage coefficients
arma¬
in the direct axis direction
to calculate the coefficients adD,
out of the
time constant
when the
armature current
Xq
required
=
reactance is obtained
—
:
per-unit
constant
quadrature
maximum,
per-unit
It is
winding,
is obtained when the armature
reactance
-
ture current is
x,
minimum
minimum,
Xd
The
a
winding.
synchronous
per-unit
xd
modulated, attaining
is
to be in line with the axis of that
maximum value when the field axis is in
a
The direct axis
current is
winding
comes
T'd.
knowledge
<rdf, fif, fiD and the time
of the reactances xd,
x'd, x"d, xt and the
The method of measurement of these reactances and
the time constant
For the sake of
ficients,
or
it
that the
was
T'd will be given in section 4.5.
simplicity in deriving expressions
for these
assumed that the three coils d, D, f have
frequency is
with the inductances
so
[3].
high that their effect will
Now at sudden short
be
leakage coef¬
resistances,
no
neglected compared
circuit,
we
have:
Va=x'd-iri
After
a
winding
i'D
while,
the
damper
is still short
coils
are
assumed to be open while the field
circuited, i.e.,
=0
then,
37
xd
'
ld
xd' ld + lf
—
0
Xd(l-<Tdf)id + if
=
substituting, {\-odf)
and
/ij
=
(1-%) (1-/0.
=
—
Xd
then
by eliminating j£,
and
dividing by ird,
we
obtain the
following
for¬
mula for iij-:
/V
=
Xd
—
Xi
above, just
As mentioned
we
(49)
—
at the moment after the sudden short
circuit,
have:
v". ;r
•*<!
—
—
'<»
v-
.
*d
j-r
j.
;r
ld
t
'd
;r
>/
j_
"t"
0
=
X,(l-ff,y)- E + U-P/)- »D+«/
o
=
x„ (i
-
ff„) /;+irD+(i
•
substituting (1 -ff<,D)
=
-
/la) rf
(1 ~nd) (1 -^D),
•
and
(x'd-Xi)
flf
and
(Xj-X,)
eliminating if, irD, ird,
we
get:
(x'l-xdixj-x,)
""
=
[x,2 xi-xd-2-^-^+x2W-xl()]
+
As for adf, adD, they can be calculated according to equations
(48) since \id, nf, fiD are now known as a function of x,, xd, x'd,
adf
=
<rdD
=
(47)
and
x"d, thus,
Hf+Hd-VdVf
(51)
HD+l*d-HdHD
(52)
As for the time constant
transient time constant
TrD, it is possible
Td
TrD (crdD),
to prove that
[3]; the sub-
=
hence,
rD
=
±L
<IdT>
=
—-l±
V-D + V-d-V-DV-d
(53)
knowing the reactances xt, xd, x'd, x'd and the time constant T'd, and
applying the equations (49)... (53) we should be able to calculate the dif¬
ferent leakage coefficients. The experimental determination of these quan¬
Thus
tities will be
38
given in the
next
chapter.
4.5 Direct axis transient and subtransient reactances and time constants
The machine
cited
so
was
driven at rated
that the output
A sudden short circuit
The
resulting
currents
speed (1500 rev/min), the field was ex¬
phase voltage is equal to the rated voltage [7].
was
are
made
on
recorded
the three
by
phases instantaneously.
"bifilar
a
oscillograph"
and
are
shown in Fig. 8.
The
peaks
and the
a
are
a.c.
semi-log.
determined from the
oscillogram,
component calculated. This
paper
sustained value
a.c.
the median line
component
was
drawn,
plotted
on
(shown
in Fig. 9). Also the a.c. component minus the
plotted in the same figure (curve C). The two curves
was
then extrapolated to the point of zero time. It is clear that curve C
composed of a straight line and a curve which extends for some cycles
beginning at the point of zero time. The difference E between the curve C
and the extention of the straight part is again plotted on the semi-log.
paper, being itself a straight line.
The per-unit direct axis transient reactance x'd is,
are
is
per-unit
open circuit
voltage of the machine immediately
before the
xd
per-unit
current from transient
component plus sustained
value at t
*;
The
=
open circuit
voltage
x'd
is
of the machine
before the
immediately
s.c.
=
per-unit
x"d
=
current from the
a.c.
(Fig. 9)
component
curve
at /
=
0
(see Fig. 9)
0.0841
The direct axis subtransient time constant
E
0
axis subtransient reactance
per-unit
curve
=
(see Fig. 9)
0.1193
per-unit direct
xd
s.c.
=
T"d
is the time
required
for the
to decrease to 0.368 of its initial value.
7.379 per unit.
T"d was found to be equal to 23.5 ms, i.e. T"d
Knowing the reactances xd, xd, x'd, xt and the subtransient time constant
T"d, and applying the equations (49)... (53) led to the calculation of the
following coefficients:
=
Hi =0.1125
adD
=
0.1308
Hf
=
0.1049
<rdf
=
0.2056
HD
=
0.0206
T'D
=
56.4196
39
o
0^
<V
1
4
3
2
flowing
current
windings
resulting from the sudden appli¬
winding
No. 3
phase
in the field
flowing
No. 2
in
=
No. I
phase
in
the stator
flowing
on
stator currents
flowing in phase
current
current
current
short circuit
three-phase
=
=
=
a
The
cation of
Fig.
The 4
digits
after the
ment is up to that
inaccuracy
that
point do not mean that the accuracy of the measure¬
position, but it is maintained in order to avoid any
can
result from the calculations.
^ I, (pet-unit)
ibo
Fig.
9
Analysis of the
a.c.
150
ms
component of the short circuit
A
=
a.c.
B
=
transient component plus sustained value
a.c.
component
C
=
D
=
transient component
E
=
subtransient component
ox
=
U\
component minus sustained value
U\
=
—-
current
1
=
—
V\
1
Xj
Xj
phase voltage just before
the short circuit
41
4.6 Determination
quadrature
Now it is
the
of the leakage coefficient
and time constant in the
axis direction
required
to
determine and
measure
quadrature axis direction. The machine is
the
to be
leakage coefficient in
manually rotated until
the axis of phase No. 1 becomes
in
Fig.
perpendicular to the rotor axis as shown
voltage was suddenly applied to phase 1, and
by a "bifilar oscillograph". Hence we have two
10. A reduced d.c.
the current
was
recorded
magnetically coupled coils, one of which is short circuited. Applying a
d.c. voltage Ud on the stator winding No. 1, the voltage equations will be:
dij/,
o.-k r1+-a
(54)
0
(55)
Fig. 10
=R0-/0+
dt
application of a d.c. voltage Ud
quadrature with respect to it
Sudden
being
in
1,2,3
=
directions of the
d
=
direct axis
a
=
=
h
in
=
=
phase No. 1, the
roto
windings
quadrature axis
d.c. voltage suddenly applied to the stator winding No. 1
current flowing in phase No. 1
current flowing in the damper winding in the quadrature
direction
42
stator
on
axis
And the flux
each of the two coils will be
linking
given by:
^i=Lqq-i1+LqQ-iQ
(56)
^Q
(57)
=
LQt-h+LQQ-iQ
Applying the Laplace-Transform
for ii(p), we get:
TqoLq
to the
equations (54)... (57)
and
solving
(P +2ap+co0)
P
where,
Laa + R'T.-
2a
=
rptt
1
It
R
2
rptt
in
Resolving
h(p)
=
J
It
-hto.
L"-T
partial
-
P
An
J
qo'^q
+ ~,
r
*
—
=
rptt
1
q
r ft
is
Uq
given by:
—
=
rptt
l
90
Lin'Rq
-—-
j
it
*-q
1
,.„
sum
\
j
+p2)
/
TI
^QQ
of these two roots (pt
«
(59a)
=
T
since the
we assume:
(P-P2)
—"««
«
^
+ ;
(P-Pi)
1
p.
-
fractions to get the time response,
approximate pole for/?!
*
T"
and
=
O
——v
j,
Tq0'L1
Lq
then,
/>2--^=~7
1-q
Thus it is
A
*i
possible
(59b)
T2
to calculate
=^
A, Bu B2.
,
(59c)
(59d)
Lqq' R—Lgq' RQ
43
B2*
(LQQ-R—Lqq-RQ)
Hence the inverse
Laplace-Transform
was
calculated and found to be:
t_
I
ilKA + Bi-e~Zl+B2-e~X2
Bu B2
where xu t2, A,
An
of this
oscillogram
with
"bifilar
a
are
(60)
given by
curve
the
is shown in
oscillograph".
equations (59a)...(59e).
Fig. 11 which is photographed
It is to be noted that the transient part
possesses two time constants, one of which is very small
other.
i[
=
Accordingly,
was
plotted
it is
on a
possible to separate them. The
semi-log. paper that is shown in Fig.
that, except for the first 10
elapses
until this
straight
ms, the
curve
more
plotted
The time
on
elapsing
initial value
was
is
a
straight
12. It is clear
line. The time which
line falls to 0.368 from its initial value
to be 44 ms. The difference between
the
compared to the
current UJR—ij^
same
graph.
It
i[
and the
was
in order that this last
straight
line
also found to be
straight
measured and found to be 4.2
was
found
portion is
a
once
straight line.
line falls to 0.368 of its
ms.
Equation (59b) enables us to calculate L'q since R and t2 are known. With
help of U'q, Lqi and Tj; T"to can be calculated according to equation
(59a).
the
1}
——
jT;0
aq
44
was
was
=
1
—
aq
can
be calculated using
found to be 0.155 s; thus
found to be 0.2825.
TrQ
equation (59d).
Wn'L^
=
=
Rq
<an-T"qa
=
48.67
NSJVWA/WWW
•
>
11
to
1
=
current
2
=
timing
resulting
current
the
phase
flowing
wave
=
10
12
t
,-Vi.
V
«'
L?2A.
Stator
voltage Ud
Fig.
*»i
»t»
.
J
Fig.
'
i
*
*
\j
in
from the sudden
application of
a
d.c.
No. 1
phase
No. I
kHz
I
15
20
Representation
25
30
35
40
of the stator current of
44
Fig.
45
11
50
on
a
semi-log.
scale
1
=
i\
2
=
the
3
=
R
line
straight
subtransient
portion
of the current
by
after the attenuation of the
phenomenon
the difference between the current
denoted
/,'
curve
/,'
and the
straight
line
portion
2
45
*.,l
•
*
•
f
«
«
Leer
-
Vide
-
Empty
5. Machine
As
as a synchronous generator
preliminary check,
the
of the short circuited
equations
synchronous
analog computer of
the Donner type. The analog scheme is given in Fig. 14 which contains
22 amplifiers, 8 multipliers and 19 potentiometers.
A curve showing the relation between armature current and field current
a
generator
—as
written,
were
and then simulated
on
an
obtained from the computer—was
For the sake of
plotted.
corresponding curve—taken from
plotted. Fig. 13 shows these two curves.
comparison,
actual machine—was also
the
the
if
•/.
/
120
110
100
90
r
80
70
GO
50
r
40
r
30
20
10-
1—
0
10
Fig.
13
20
Steady
1
t
60
70
'
30
40
50
if
—
1
^
80%
state short circuit characteristics
O
=
from the machine
o
=
from the computer
47
The
of the
voltage equations
0
-0.0571
=
irx+
three-phase
short circuited generator
dfc
are:
(61)
—
ax
0
-0.0571
=
ir„+ -^
(62)
ax
Ur,
irf+116.2-^
(63)
=irD+56.42-^
164)
=
ax
0
ax
dVo
0
=f+48.90-^
(65)
.
ax
linkage equations
The flux
of the different coils of this generator
are
given by:
i'd+irf + irD
(66)
0.4611
ft+ff+0.8951 fD
(67)
0.5045
ij+0.9794 /;+fD
(68)
^=
0.2684
iJ+rQ
(69)
yQ
0.1926
/,'+$
(70)
M
=0.5805
\jjrf
=
\l>rD
=
=
The transformation
\l/rd=
Vq
=
i£
=
i^
=
~
equations
are:
t/^-cosy + i/^-siny
(71)
V*
y
(72)
-cosy-/£ -siny
(73)
*,
si" V +
tiff
ird -siny+ij
cos
(74)
-cosy
equations can be simulated on the analog computer, we have
to introduce the suitable amplitude and time scales. As for the amplitude
scale, it was assumed that every 100 V correspond to 100% of the reference
value of each quantity, e.g. V's s 100 V, when U}
100% of its refer¬
Before these
=
ence
value.
As for the time
time w.r.t. the
t
48
=
5 /„
scale, the relation between the per-unit time
analog computer tc
was
x
and the
chosen according to the relation:
(75)
-0,5 K
>v,
I
0.51?
Jk5^
7pMi^t
Fig. 14
Analog computer scheme for
generator driven at constant
the short circuited
synchronous
speed
AAA
=
,42.1
,44.2
=,410.1
,410.2
=
,42.2
A5.3
=
,413.1
=
,418.1
,420.3
=
;r
5.1
=
;t
A2l.2
=
n4A
A21.3
=
7t6.1
0.1722
,48.1
=
,417.1
AY2A
=
,47.1
,420.1 =7tl.l
,420.2
=
7t
,420.4
,421.1 =7t2.1
,419.1
7.1
,421.4
=
?r
3.1
8.1
Pl
=0.3443
P2
=
P3
=0.4307
PA
=0.9794
P5
=0.5045
P6
=0.4307
p1
=0.4431
Ps
=0.4475
P9
=0.4307
P10
=
0.5709
Pn =0.7452
P12
=
0.5112
P13
=
0.7452
P14
^,5
=
0.5709
P16
=
0.0040
P18
=
0.5
corm
s^
=
50 sin y
sz
=
50
s3
=
—
cos
50
=
0.5
=
0.1926
P17
=
0.5co;
P19
=
0.4611
=
0.5
y
cos
y
49
Leer
-
Vide
-
Empty
This
means
machine / and the time
tc
=
synchronous
analog computer tc is given by:
that the relation between the time
the
on
on
the actual
20 nt
speed of the synchronous generator was assumed to be constant and
equal to that of synchronism. Hence the sin y and cos y were obtained
by solving the differential equation:
The
-£--25,
(76)
analog set-up for solving this equation is done by means
integrators .420 and A21 and inverter .422 of Fig. 14.
The
of the two
amplifiers, there is
a tendency
slightly
amplitude. With
the course of time this change in amplitude can be dangerous. To com¬
pensate this effect, a part of the output of integrator .420 is given to the
Due to the very small
encountered in the
phase shift
for the sine
wave
to be decreased
of the potentiometer PI6 and an
potentiometer P16 adjusted to the
having an amplitude of 50 V has decreased
amplifier ,422, by
input
of the
input
resistance of 4 MQ. With the
value of 0.004, the sine
wave
means
of 10 minutes. The results were,
period
3 minutes of computation.
Introducing the mentioned
by
1 V in
a
time scale in the
making the necessary alterations,
we
ready for the simulation
the
are
in
-^
=
-ft
=
-0.5
ft
fa
yQ
-
-
on
however, taken within
equations (61)...(74), and
following equations which
obtain the
analog computer:
J0.2854i'xdtc
(77)
j 0.2854% dtc
(78)
=
-0.0215
J* {irt-ft)dte
(79)
=
-0.0886
j irDdtc
(80)
=-0.1022
j rQdte
(81)
-Vd
=
-(^-cosy + ^-siny)
(82)
-Vq
=
-(-^-siny+^-cosy)
(83)
(fD + ff
(84)
0.25
0.5
ird
ff
W
=
-
0.4307
=
-
(0.2306 ird+0.4475 irB
-
-
0.5400
i/r})
(85)
51
=
(87)
=-(0.1926 i'q-if,'Q)
(88)
=
-
—
.
0.5
ir.
(86)
(irQ-yq)
=
0.5
-(0.5045 ij+0.9794 J}-«^>)
=
3.7261
(0.5 iTq
•
sin y
—
0.5
irt
cos
y)
(89)
—(—0.5 /^-cosy-0.5 j^-siny)
(90)
The
analog scheme representing these equations is shown in Fig. 14. U}
was
magnitude, then the amplitude of the field current i} and
the /£ were measured. Since /£
0, as the machine is connected in star,
varied in
=
then ig
if
=
i\.
The armature current
and is shown in
with the calculated
The
52
following
i\ was plotted against the
field current
Fig. 13. This shows that the experimental results
agree
ones.
values
are
valid for
Fig. 14, 17, 19, 21, 24, 25, 32 and 34.
Rt
=
100 kQ
R7
=
800 kQ
Rl3
=
10 kQ
R2
=
200 kQ
R8
=
1 MQ
R1A.
=
15 kQ
R3
=
300 kQ
^9
=2MQ
Ct
=
0.1 uF
J?4
=
400 kQ
R10
=
4 MQ
C2
=0.2uF
Rs
=
500 kQ
Rlt
=
5 MQ
C3
=ljiF
R6
=
600 kQ
R12
=
10 MQ
C4
=2uF
Analog
6.
computation of the single-phase controlled synchro¬
nous MACHINE
6.1 Mathematical formulation
investigate
The aim of this
chapter
only
of the stator of the
one
phase
d.c. power
is to
supply through
4
the
case
shown in
Fig. 1, where
synchronous machine is fed
thyratrons.
from
The calculation will be done
a
by
expression for the voltage drop across the
Integrating this voltage over the period from
smoothing
a' to fi' and from (a' + rc) to (fi' + n) (where a' is the ignition angle and
f}' the extinction angle), and forcing the input of the integrator to be zero
in the period from /?' to (a' + n) and from (/?' + n) to (a'+2 7t), the thyratron
obtaining
a
mathematical
inductance Ld.
action is thus realized.
Fig. 15a represents the equivalent circuit of the
ting thyratrons 1 and
stator when current flows
voltage drop across the conduc¬
Neglecting
2, then the voltage drop across the smoothing in¬
in the forward direction.
the
ductance is the difference between the
applied d.c. voltage Ud and the sum
voltage from the synchronous machine in the phase No. 1
voltage drop across the stator resistance, i.e.
of the induced
plus
the
^-"-(«-^)
Dividing by
the reference value of the
form it into the relative
form,
we
voltage Uny/2
in order to trans¬
get:
*%-^(*-*+%)
where the
per-unit value
of the
smoothing inductance
is
cqb-L„
and the
per-unit
Introducing
value of the
the time
applied d.c. voltage is
scaling mentioned in equation (75),
we
get:
53
v
(a)
(b)
Representation
Fig. 15
of the stator of the
single-phase
controlled syn¬
chronous machine
=
the
Ld
=
the
urL
=
Ud
per-unit value of the d.c. voltage driving the stator winding
per-unit value of the smoothing inductance
per-unit value of the voltage drop across the smoothing in¬
the
ductance
u\
per-unit value
the
=
~R
h +
i[
=
the
Rr
=
the
the
stator current
stator resistance
per-unit value of the total flux linking
per-unit value of the time t
\\i\
=
=
the
=
case
=
phase voltage
~dT
per-unit value of the
per-unit value of the
t
(a)
(b)
of the
case
the stator
winding
when current flows in the forward direction
when current flows in the
reverse
direction
thUsn=-Lj(5R^n+d^-5urd)dtc
The
integrated function should be" available
for the
period
from a! to B'. It should be
at
the input of the
integrator
period
from B' to
zero
in the
n+a'.
Now
considering the
case
in the
period
Here the direction of the current
the
54
where
thyratrons
assumed to be the
(a'+7t). This is illustrated
the period (7t+a') to (rc+/?'):
period
Thus for
was
from a' to
in
3 and 4 conduct.
same as
Fig. 15b.
that for
since /d-c-
then:
—iu
=
di,
Introducing
dij/A
I
the
same
amplitude
given before,
and time scales
we
get:
—^J(5jr-r'+7^+S0S)*-
r'
voltage that equals Urd during the period from a' to
Urd during the period from (rc+a') to (2n+a'), and
(jt+a') and equals
if we denote this voltage by ura, we obtain an equation valid for the current
i[, so long as the integrated function is available during the conduction
periods and so long as that function is made zero during the nonconduct¬
Thus if
we
form
a
—
ing periods.
The equation
of the stator current is thus:
of the conditions made
practical realization
The
on
this
equation
so
that
in section 6.2.
explained
following equations, together with equation (91) describe the pheno¬
menon appearing in the single-phase controlled synchronous machine:
the
effect may be fulfilled will be
thyratron
The
-0.5
¥t
-0.25
-5^
0.5
fD
2*a
0.5
i'f
0.1^
J (Urs-i'f)dtc
-0.0217
J- i'Ddtc
-0.5112J- irQdtc
(79)
=
-(-0.5 1/^+0.4897 /}+0.2523 i\ -cosy)
(94)
=
-(-2^-0.3851 i\-
(95)
=
-(-0.5^+0.2306h -cosy+0.4475Q
=
-[-(0.0424+0.0156cos2v)-i'1
=
Vd,=
=
-0.0215
(92)
(93)
sin y)
(96)
-0.1(/}+Q-cosy+0.1 irQ-smy]
It is the
general practice in
differentiation since it is
some cases
unavoidable
d^\ldtc
number of multipliers
tion
a
as
the
noise
in
analog computing technique
amplifying
our case
quantity \j/\,
as will be given
out of the
(97)
where
to avoid
process. However, it is in
we
otherwise
have to form the func¬
we
in details in
need
a
much greater
chapter
9.
55
Satisfactory results can be obtained by approximate differentiation which
can be made to approach arbitrarily close to the true derivative. Compro¬
mise has to be done
the
based
that the noise is not too
the solution of the
on
z
so
high and
at
the
same
time
in differentiation is not great. The circuit for differentiation is
error
jz-dt
=
implicit equation:
+ k-z
3
I
—.
s5
|
*
°
4
s6
it s4
5
Fig.
16
diagram of the thyratron simulator for
synchronous machine
Block
controlled
1
=
Schmitt
2
=
monostable multivibrator
trigger
trigger
3
=
Schmitt
4
=
gate
5
=
clipper
6
=
bistable multivibrator
=
summing amplifier
st
=
50 sin y
s2
=
reference
s3
=
ignition pulses
s4
=
cut-off
pulses
s5
=
square
wave
s6
=
output of the gate
j7
=
7
'
U0
"56
=
square
d.c.
the
and differentiator
pulses
wave
voltage
controlling
=
to
the gate 4
output of the bistable multivibrator
compensate the d.c. level of s7
single-phase
Rt
(?o
os u;il
Rio
p.
K.
0.5if
tftt
6V*W-6VS4
Analog computer
synchronous machine
Fig. 17
scheme
simulating
,42.1
=7i
3.4
,42.2
=
A 6.2
=tt3.3
A9.1
=A20.l
the
,46.3
single-phase
controlled
,42.3
=,46.1
,49.2
=kS
,410.1 =k9
,416.1 =tc6.2
A\6.2
=
n5.\
A16.3
=
tc
2.2
A 17.1 =7t3.2
,417.2
=
tt1.2
A 19.1
=
7i
6.3
^19.2
Pt
=0.4304
P4
=0.4475
p7
=0.5046
P10
=
variable
P13
=
0.5
Pis
=
0.1972
P18
=
0.5804
Ji
<
=
0.5000
=
j3
=
j4
=
100
C/0
7*51
7:1.3
P2
=0.3443
P3
=0.9224
p5
=0.4431
P6
=0.9794
P8
=0.5112
P9
=0.7702
Pn
=0.9980
Pi2
=
0.5709
P16
=
0.8000
P14
=
0.5
P19
=
0.0040
P17
=
0.6240
=0.25(i} + ^)
s6
j2
=
0.1L
s5
=
50
<
=
0.5000
r^i
(0.3120 sin2y-0.5804)
50 sin y
cos
y
57
=
=
=
—
50
Uo
=
cos
y
100 V
thyratron simulator
j7
=
-0.1
dtc
Leer
-
Vide
-
Empty
which
gives:
P
\+(l-k)-pX
=
so
that
dx
urn z
=
dt
a-.i
This is done
by
means
of the 2
amplifiers A2Q and ^421 and the integrator
Fig. 17. The potentiometer-setting k is ad¬
.422 of the circuit shown in
justed
the
as near
unity
building
also used, namely by
6.2 Simulation
permits which achieved in our
speed was assumed to be constant,
the noise level
as
0.998. Since the
=
method of
same
was
to
the value of k
case
of the
the sin y and
solving
inverter
A transistorized circuit
was
cos
y
the differential
explained in section
5
equation (76).
effect
formerly built in a dissertation made in the
Engineering (Diss. No. 3060, ETH, Badr)
Institute of General Electrical
required in this case. Benefit was made
adding the necessary completions to fulfil
our problem. A block diagram representing
to obtain a function similar to that
of a
the
part of this apparatus after
requirements asked for in
one
phase of the mentioned circuit
In this
tion
diagram,
periods,
zero.
The
reference
vibrator.
in
Fig. 16.
intentionally made equal to
ignition points are obtained by delaying the
pulses defining the
pulses by the angle a.'. This is done by a monostable multi¬
The cut-oif pulses are obtained by means of a Schmitt trigger
In order to
generated.
it
crosses
complete
ments of this
zero
so
problem, the
The series of
were
the
the circuit
made to
axis.
that it may
voltage ura
pulses coming
comply
a
constant
trigger
voltage U0
amplifier A 8 of Fig. 17.
of ura [equation (91)].
In order to vary the
to
a
potentiometer.
(a' + k),
a
require¬
out
of the monostable multi¬
bistable multivibrator. The d.c.
to it. This takes
was
place
com¬
eliminated
in the
The output thus obtained fulfils the
by
summing
requirements
amplitude of ura,
the output of amplifier A8 is fed
positive during the period from a' to
negative during the period from (n+a.') to (a'+2a).
Thus
and it will be
with the
mentioned in section 6.1 should be
ponent contained in the output of that multivibrator
adding
during the conduc¬
otherwise this function will be
when the current
vibrator
given
is
the gate allows any function to exist
ura
will be
59
6.3
Analog
set-up
(97) describing the phenomenon
The equations (79) and (91)
ulated
on
nineteen
the
analog computer Twenty-two amplitiers,
were
multipliers
sim¬
and
were necessary to carry out this simulation
potentiometers
The transistorized circuit whose block
used
si\
diagram
is
shown
in
Fig
16
was
conjunction with the analog computer (of the Donner t\pe)
in
fulfil the ignition and extinction conditions
requned
because ot the
to
thy-
ratron effect
The
analog
scheme
meters on the
is
shown
in
required values,
consideration the
loading
Analog computet tesults
6 4 1
Oscillograms
to
take into
=
18
Fig
100 V
=
=
to
Oscillograms
5",, of the reference voltage The angle
of the
single-phase system
100",, ot the reference value
V}=
30",,
m~-
100",
(,;„;
=1
The
IHj
1=05
u\
y.
=
140
i/VWWt/VV
~
60
used
oscillograms representing the case of the single-phase con¬
140° and for an
synchronous machine for a firing angle of a'
applied voltage equal
Fit;
was
18 shows
trolled
U'd
the null voltmeter
effect of the next stage.
6 4
Fig
Fig 17 For the ad]ustmentof the potentio¬
-
t
no
load
phase voltage u\
y
was
_5oy_
A
J
__
50V
Fig.
18b
I
0.5
=
The
u\
phase voltage u\ and
2
=
the
phase
current
/[
0.5 /:
5V
50V
,£v-
/•"/#.
i'n
The
I fie
and the
=
0.25
-wvwvwav
phase
/,;
2
50V
=
0.5
damper winding
in the direct axis direction
i\
i\
WvVvWvyvw>
Fig. I8cl
i'q
current
«1
5V
tion
in the
current
The current in the
and the
1=2/'
phase
2
=
damper winding
current
in the
quadrature
axis direc¬
/[
0.5 /T
61
20V
50V
Fig.
lHe
i
0.5
=
The field current
/;
2
=
i\
and the
phase
current
i\
/;
o.5
1-1
sin2T
k
Fig.
v;— I
19
=
Figure illustrating the definition of
axis of the
phase
between the a\is of
1
=
the
I
=
pulse which determines
=
the
i!
angle
firing angle
the
phase
beginning
No. 1 and the
phase
point
of
begin of
sin •/, i.e. it is the
No. 1 and the rotor axis at the
phase. This is illustrated
duction in that
fundamental
frequency equal
is valid for both
note
62
axis
phase
No. 1
in
Fig.
between the
of
of
begin
con¬
19.
damper winding
has
to double that of the stator current. This
damper windings.
the effect of the
angle
point
It is to be noted that the current that flows in the
a
rotor
of the current in
firing angle
measured from the
a.xis of the
the
No. 1
The field current is also
stator current on
it.
formerly
given
and
noted in Fig. 2.
we
6.4.2 External characteristics
i\
The average value of the stator current
to the
is to be determined
according
equation:
j.
and since the
period Tc
1.2566 s, then
=
Is.
2
J'c-
0.6283
J
l
dtr
o
Tc
2
=
1.5916^ i\dtc
Thus if the current
put multiplied by
i\ is integrated
over
the factor 1.5916,
we
half the
get
d.c. current taken from the d.c. power
achieved with the
of the
integrator
Fig. 20
help
of
a
every 0.5
Analog scheme
cyclic
Tc
a
period and
then the out¬
voltage proportional
supply.
The definite
integral
reset generator which resets the
s, i.e. after 0.6283
s.
to the
output
Since the cyclic
to calculate the d.c. current
is
reset
ia.c. in the single-
phase system
P
=
0.6366
S1
=
a
switch activated
by the cyclic
reset
generator every 0.5 Tc
63
generator possesses
0.7093
effect
reset time of 0.081 s, then its interval should be
a
cyclic reset generator has no harmful
anticipated since the period of integration will
the moment at which it begins will be different.
The finite reset time of the
s.
on
the results to be
always be
Tc, but
0.5
The circuit used to obtain the d.c. current is shown in
put of the
used in this circuit
integrator
was
Fig. 20.
The out¬
recorded with the
help
of
mingograph 230 recording instrument, being a 4-channel recorder driven
a synchronous motor.
The voltage ura was also recorded using the same recording instrument on
another channel. For every firing angle a', records were taken for /,J-C.
a
by
ura was
angle a'
where
The
and
volts until the
"tilting-limit"
of Fig. 3
(the
results,
one
of the
replotted
on
curves
corresponding
the same figure.
to the
curve
voltage drop
across
the
tween 15...20 V. Since in
time, then
drop
and
a
across
shifting
Fig. 21
nous
1\
our case
these two
the
curve
V
one
at the
using
a
accuracy up to 0.1
firing angle a'
no
=
130°) was chosen
provision
was
tubes which ranges
two
was
an
thyratrons
assumed
thyratrons. Converting
by this value, we get the
region
External characteristics of the
as
conduct at the
the
sum
this to the
dotted
when
made for
normally
£ is
single-phase
of the
voltage
per-unit value
curve
near
which coin¬
180°. At values
controlled
synchro¬
voltage (per unit)
=
d.c.
=
d.c. current
(per unit)
c
1
=
firing angle
a'
=
170°
=
firing angle
a'
=
160°
3
=
firing angle a'= 150°
4
=
firing angle
a'
=
140°
5
=
firing angle
a'
=
130°
6
=
7
=
120°
firing angle a!
corresponding curve
=
the
machine for a'
=
130°
taken
directly
from the
be¬
same
machine
2
64
thyratron
voltage drop of 35
cides with the calculated
Urd
achieved.
time
of the external characteristics
It is to be noted that in the calculated results
the
was
measuring
ms).
measurements taken for i^c, j/a with a! as parameter were plotted
shown in Fig. 21. For the sake of comparison between experimental
was
and calculated
and
zero
measured with respect to sin y
[20] (being of the digital type with
instrument
The
varied from
synchronous
10
15
20
8
=
conduction angle £
=
80°
9
=
conduction angle £
=
100°
10
=
conduction angle £
=
120°
11
=
conduction angle f
=
140°
12
=
conduction angle ^
=
160°
13
=
conduction angle f
=
180°
25
30
35
65
of
£ far from 180° electrical, there is
and
of
experimental
Fig. 16 is
input
to the
voltage drop
in the
integrator
across
integrator
deviation between the calculated
A10 of
resulting
Fig. 17 has
larger
was
in
to be zero, there is
voltage drop
nonconducting period
constructing
on
improving
this circuit
is
the circuit
thyratron effect in the three-phase case which will
Another method for
since the gate
is
a
small
integrated
increase in the output d.c. current,
an
when the
remedied
expected
that during the interval where the
sense
the diode of this gate. This
A10
and this increase is
ever, this effect
in the
perfect
not
a
This deviation is to be
curves.
was
be
given
given by
larger. How¬
simulating
the
in section 7.3.7.
Dr. Badr in the
"locomotive session" held in Zurich in Oktober 1961.
6.4.3 Current locus
It is to be noted that both the
phase
harmonics. The current
be put in the form:
'i
i\
can
current and
x
=
and it is
u\
contain
I\c- cosx+lric- cosS x+Irlc- cosl x-\—
=
+7rls-sinx-|-/5S-sin5x + /7S-sin7x +
where
phase voltage
y
=
5
Vle
+
(98)
a>rm tc,
possible
=
---
to
put the voltage u[ in the form:
cos x
•
+
VSc
cos
5
x
+
Vlc
•
cos
7
x
-\
(/rls-sinx+[/5S-sin5x+l/7s-sin7x+---
(99)
In order to determine the behaviour of the fundamental component of
the current, it is necessary to determine the value of
These values
determined
were
according
Tc
I[c
=—
J
to the
I[c, I[s, U[c, XJ[S.
following equations:
1.2566
i'i-cosx-dtc
=
f
1.5916
co
i\-cosx-dtc
(100)
o
1.2566
7rls
=1.5916
j
o
i[-sin x-dtc
(101)
u\-cosx-dtc
(102)
m1-sin W/c
(103)
'
1.2566
V'lc
=
1.5916
j
o
1.2566
Vu
66
=
1.5916
|
\
o-
T
T
V
/\
/ \
zx
=
50
sin?
z2
=
50
cos
z3
=
0.5
i\
z.
=
0.1
z6
=
0.6283
=
0.6283
=
0.6283
=
0.6283
a
I[,
/fc
U[c
U[s
period
switch in
position
I
position
II
switch in
position
position
I
switch in
St
and
of 1.2566
S2
are
x.
of these
amplitude
voltage
Hence it is
by the cyclic
reset
generator
of the fundamental components of both the
and their
possible
activated
equations is fulfilled by the scheme given in
quantities I[c, I[s, U[c, U[s enables the
analog set-up
22. The determination of the
calculation of the
II
s
The
sin
the current locus
switch in
Fig.
current and
2,
s~
y
The two switches
having
12
Analog set-up for determining
Fig. 22
z5
-^H
*—
z2o
phase angles
to determine the
with respect to the function
amplitude
and
phase of
the
fundamental component of the current with respect to the fundamental
component of the voltage. This
was
executed for different values of a'
parameter and the resulting family of
curves
is shown in
as
Fig. 23.
67
Re(I,)%
Current loci for the
Fig. 23
I±
=
complex
single-phase operation
vector of the fundamental
component of the phase
current
Refjj)
=
the part of the fundamental component of the
which is in
phase
phase
current
with the fundamental component of the
voltage
/w(7j)
=
the part of the fundamental component of the
which is in
quadrature
phase
current
with the fundamental component of the
voltage
1 : a'
=
170°
3 : a'
=
150°
5 : a'
=
130°
2 : a!
=
160°
4 : a'
=
140°
6 : a'
=
120°
The first
in
phase
works
the
68
quadrant,
where the fundamental component of the current is
with that of the
as a
region
motor. The
voltage represents the region where the machine
quadrant
where the machine is
where this component is in
a
generator.
antiphase
is
7.
Analog
computation of the three-phase controlled synchro¬
nous MACHINE
7.1
The mathematical formulation
Feeding the stator of the three-phase synchronous machine from a d.c.
power supply through 6 thyratrons, connected in a bridge circuit, results
in stator currents represented in Fig. 24. The current in each phase suffers
6 switching intervals for every 180° electrical. In order to be able to
simulate these currents on the analog computer, it is necessary to obtain
a mathematical expression for the derivative of the current flowing in
each
phase. This expression should be valid during
that
carries current and should be
phase
Considering
made
the current
during
during
flowing
in
phase
the intervals AA' and
be valid
zero
The fact that the
zero
No.
BB',
the interval at which
outside that interval.
1, this expression should
and should be
intentionally
the intervals A'B and B'A.
current suffers 6 switching actions during each
dividing this period into 6 intervals, let us denote
phase
180° electrical suggests
by the letters a, b, c, d, e,fas shown in Fig. 24. The second half period
(180°...360°) was also divided into 6 intervals, denoted by the letters g,
h, i, j, k, I. During the overlapping interval a for example, the phases 1
them
and 3
are
power
parallel and their combination is in series with
smoothing inductance Ld. They form the load of the d.c.
connected in
2 and the
phase
supply Ud.
Applying
Kirchhoff's law of
the inner
loop consisting
Va
thus
Ld
'
=
Li
dit
—
Ud
=
dt
2
+
Substituting
(il
1
+
of
circulating
phases
currents
1 and 3 in
on
this
parallel,
we
loop,
and
on
get:
i3) + R{il-i2)+—{xl>l-il,2)
d2
Ld
2R
•—rW^W3
dt2
—
^t)
VV
2(i2"''l)+2"^W2"W
the values of
i/^, ij/2, \j/3 given in
{104)
the
equations (29)... (31)
in
69
'1
/
A 1
\
i
\«
d
*
\!
i\-
f
lei
ii
ii
j
|k|
y«
•
y
f
f
t
!/b
i/i
:!
1
i
"
iUUi
f
.
!
i
i
ii
ii n irm ii ii,
*
i
!
i
ii
ii
!
i!
II
II
1
i
i
=
b, d,f, h,j, I
=
the currents
equation (104)
'
dt
overlapping
nonoverlapping
and
taking
2R
pole
the
d
=
=
the stator of the
\2V*
(
=
0,
we
get:
V"
2
J3
3
\j/0
\
(1°5)
2(/2-/l) ^(-4^+V')
+
phase
dit
'IT
flowing in
machine
nonoverlapping period b, the
supply through
1 and
,
intervals
of the d.c. power
phase
!
ii
ii
into consideration that
dt2
R
During
i
intervals
2
+
i
i
ii
three-phase controlled synchronous
a, c, e, g, i, k
i
on
i—-H—r i
Diagram representing
Fig. 24
70
j'!
h
a<'
!
]ci
b
i
i3
\jx n!gi
ii
r
ai
'2-
i i
2.
Applying
Ud + R(i2-il) +
current flows from the
the
smoothing
Kirchhoff's law,
—
dt
Ud + R-(i2-il) +
we
positive
inductance
Ld,
get:
(il,2-^1)
dt
-|*-+X*'
(106)
If
continue
we
using
we
the
get
This
same
this
expression for Lrd (di[/dtc).
an
was
procedure on the rest of the intervals, and
amplitude and time scaling formerly given in chapter 5,
applying
carried out, and the results
shown in table I.
are
Table I
L'd
Interval
b
—
=
dtc
5Ud
Ld
2
10Rr
Jl
d2
[3
dt2c\2Y"
5c/;+5K'ow1)+
d
'
Y'
2
l
./3
3
—l--^+Y*r
d
Jl
(3
'
c
5[7;+5^'(i3r-iD-^-l2^+^-^
d
5Ud
+
5R'&-i\)-
5Urd
Ud
2
10Rr
5Rr
/
*
d2 (
dt2c\
r
r^
'
J3
(3
d
—
'
l-V*+^-rf,
73,,'
3
2Vx
(3
d
1
Yp
2
Ir
73
/r
0
-V-^'TSU'+^J
10Rr
2
5/?*"
,
dt2A2Y*
1
d
(
rf
-5^ + 5^(^-/1)+—
-5Ud+5Rr(r3-i\)+
d
—
2
3
/
I
r
J3
3
-2^+
3
'
,/3
2
^
Ji
{--^~Y^
71
Table I
Interval
Lrd
/
0
Carrying
this
expressions
—
dtr
procedure
on
Ld (d ir2/d
for
(cont.)
the
phases 2 and 3, with the aim of obtaining
Lrd (d f3/d tc), we get the equations tabu¬
tc) and
lated in table II and table III.
Table II
rdir2
Interval
Ld
—
=
dtc
d
b
jl
(3
-5Ud +
5R'(i'1-i'2)+—\-yi-YrP
-5Ud +
5Rr(i\-Q+-^-^:-:~-rfi
d2
Ud
5U'd
.-
,
s
—T+m:^d-?y^^)
5R'.r
1
d
(3
.r
y/l.;
0
1^ + iL.A2
2
+
77^-r-
—
lORr'dt
/
>/?./.-,
3
\-^Vp+^
+?«-*>4£<-^«>
72
'
Table II
(cont.)
rdir2
Ud
Interval
f
—
dtr
5Ud+5Rr{f3-i2)+-^-{-j3r^
g
5
u:+5R'd\-i'2)+
h
5
Urd +
j- {l^-^-r,
5Rr(f1-i-2)+-^.^-:f^
5Ud
d2
Ld
r
,
5Rr
J
d
1
d2
Ld
5Ud
,
(3
/3
J~3
73
-5L/dr+5Rr(r3-r2)+^-(-73^)
Table III
rdi\
Interval
Ld
=
dt„
d2
5Ud
Vd
2
10iT
1
d
73/r\
(3
dfzr\2
,
,-
N
5R\.r
Table III (cont.)
Ld
Interval
b
5
Ud
+
d2
L'd
——
1
r
,
5Rr
,\
—vU/3iK) +
•
(3
d
Jl
T
r
O'i
-1\)
r
(,/3^)
_5i/; + 5K'(,'2_r,)+
d2
-5Ud+5R'(i'2-i'3)+-^(j3yi>)
d2
/
=
—
dtc
0
c
e
dir3
(3
L'd
5Urd
—+
g
Ld
(3
lft
y/3ir\\
J3
0
2
5Rr
VpJ
d*rv
ioj?r
d
1
r
rs
d
5
i/i+5R'(i'1-i'3)+
—
a
/3
(3
l-^+
ir
73
^-^
tc
5t7;+5^(ir2-r3)+^-(V3^)
74
-/3
,
Table IV shows
each
of the
these
quantities
during
to
currents
during which
intervals
and the
should be present. Each
quantity
corresponding interval, otherwise it
effectively done by means of
the
consists
of
number
a
of
a
the
defining
gates
equal
transistorized circuit
right intervals
in table IV. The transistorized circuit will be
specified
each of
should be present
should be made
This is
zero.
which
the different terms that constitute the derivative of
stator
explained in
detail in section 7.2.
If the
right quantities constituting the derivative of each of the stator
given to three integrators, then the outputs will represent
currents are
the three stator currents shown in
Obtaining
the three stator currents, it is
transformations
(13) and (14)
along
serve
the direct and
xj/q respectively. \prd, \j/rq, \j/a
(17), (20), (10) and (11).
using
could be
possible
quadrature
to obtain the currents
Equations (19), (21), (5) and (6)
the
Fig. 24.
irx, i'p, /J,
Equations (7), (8),
and
irq.
irD, i'Q, \}/rD and
using the equations
allow the calculation of
\j/p
can
be calculated
The derivative
principle
adjusted to the
same
and
to make the necessary
axes.
and
dtj/a/dtc
d^\dtc
was
done
mentioned in section 6.1, where the factor k
value of 0.998.
Performing the differentiation another time
tion becomes discontinuous. However, it
is
more
difficult
as
the func¬
to obtain satis¬
possible
adjusted to 0.9, but
because of the shortage in the available number of amplifiers, the
differentiation was performed using only one amplifier after shunting
it with a condenser of 0.1 uF which is effectively the same as if k
factory
0.9.
were
25 shows the
Fig.
was
results for this second differentiation with k
complete analog scheme used
to simulate the three-
phase controlled synchronous machine.
Because of the lack of
a
two-channel
work, the system was solved for the
cos
y
Mre
where
calculated by
were
calculated
according
=
M'e
multiplier
case
beginning
of the
solving equation (76).
equation:
The electric torque
was
to the
rq-ird-tt-irq
=
at the
of constant speed. The sin y and
the electric torque
(107)
resulting
from the machine in per
unit.
75
Table IV
Constituents
Current
Interval where
derivative
each constituent
exists
(1)
50 Zl
=
V*
+
25 y3
=
25
h)
+(-h
dt\
4000*'
800*
d2^
3L5
4000 #"' df2
80
+
(2)
50
80
a,g
'
df„
01-*
800
df„
R
3
2''20
dV.
'
200
dt.
20+ 200
d<c
b,h
'
200
t; dA
'
'
4
dt.
(3)
25 yi
=
+
(4)
50 z2
=
+
76
25
1.25
(6)
-1.25
0l
3)
c,
d, i, j
V3 dtf
200' dfc
50
US
3LJ
d2^
80
4000 Rr
dt2.
^\U,
d 1¥s
4000 Rr
dt*
e, k
R
l
80
2_.dJ<L J±.dJ±
800' dt, +800' dt.
Urd
(5)
dfc
Urd
h, i,j, k
a,
b,
c, d
Table IV
(cont.)
Interval where
Constituents
Current
each constituent
derivative
exists
2j.r
(1)
50 z4
=
50
4000Rr' dt\
80
^« *'« + «;-«.£
"
rf«2
4000JT
80
e,k
'
dtc
400
(2)
25 y5
=
(»3-»2)
3
2'
25
/,/
100
20
_
(3)25^
K
djz
4
dt„
=
25
0Wz)'
Rr
20
+
3
d_K
200
dtr
g,
/i,
a, 6
yfidlj^
d<„
200
(4)
50 z3
=
50
,
urd
Jin
80
2000Rr'df,
Rr
.
3
2.IS
d^\
dV.
i, c
V3 ^
800' rffc
(5)
1.25
(6)
-1.25
l/J
Urd
/,
g,
fc, /,
/'.
i
a, b
77
Table IV
(cont.)
Interval where
Constituents
Current
each constituent
derivative
exists
(1)
50 zs
=
50
V3l; d2^
'Uj
2000 Rr'
80
Rr
+
dt2
dya
3
i, c
(ll"'3)'80+800"7r
y/3 d^
800' dtc
K'
(2)
25 y,
=
25
("-l3)-20
dft
3
+
2oo-7T
rf'J
V3 ^
200'
(3)
25 y6
=
(4)
50 z6
=
+
+
25
50
-1.25
°2
3)
_Ui
80
moW
+
20
df.
100
3Lrd
d2y*
4000 Rr
dt2
7F+°2
3)
so
«,/, fc,
/
,g
V3 dj£
'
400
(5) 1.25 t/J
(6)
df.
Urd
dL
j, k, I,
c,
a
d, e,f
;ns*'o-
u
01
"*>
(Jl
Ul-*
w
a>
a
CD
1^)00
r-C»
O
q
>>
>>
>>
>>
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Empty
25
Fig.
Analog computer
scheme of the
three-phase
controlled syn¬
chronous machine
,42.3
=.410.1
.47.2
=
,418.1
,412.1
=
,412.2
=
.432.1
,415.3
=
,428.1
,416.3
=
,428.2
A 17.3
=
.434.1
.419.2
=
,4 32.2
,419.3
=
,440.1
.420.1
=
,426.3
A22.1
=
,435.2
,424.3
=
,434.2
.425.3
=
.442.2
A26A
=
A16.l
,426.5
=
,435.1
,427.1
=
.428.4
,427.2
=
,442.3
>431.1
=
,42.1
.432.3
=
.415.1
,432.4
=
,424.1
,433.1
=
,434.4
A33.2
=
.428.3
A35A
=
A42.l
,437.2
=
,47.1
.438.1
=
,434.5
,438.2
=
,442.5
,438.3
=
,428.5
.438.4
=
,435.3
,426.2
.415.2
.438.5
=
.417.2
,438.6
=
.439.1 =.442.6
,439.2
=
,434.6
.439.3
=
.428.6
=
.424.2
,439.6
=
,425.2
,439.4
=
,416.2
.439.5
,440.2
=
,417.1
,440.3
=
,442.4
,441.1 =y425.1
,441.2
=
,434.3
A42.1
=
A26.l
,442.8
=
.429.1
,444.1 =tt5.1
A44.2
=
n3.\
.444.3
=
7:1.1
,444.4
7:7.1
,445.1
=
tt6.1
,445.2
=
7:2.1
.446.1 =7:8.1
,446.2
=
7:4.1
7i
n
5.3
=
=
1.3
,437.1
=,419.1
-
pt
=0.3443
p2
=0.3443
P3
=0.9794
P4
=0.1432
Ps
=0.0886
P6
=0.2045
P7
=0.8660
Pe
=0.2000
P9
=0.4330
P10
=
0.9980
plx
=
0.4451
pl3
=
0.4451
P14
=
0.1123
P16
=
0.3936
0.6307
P19
0.4611
=
0.2840
P22
=
0.1123
P17
P20
P23
=
=
=
0.3851
P25
=
variable
=
0.1123
0.9980
P12
0.1123
^5
0.5805
P18
=0.1123
P21
0.5368
P24
0.1123
P27
P28
=
0.2840
=
0.0200
P30
P26
P29
=
=
=
=
=
-0.015^
j2
=
-0.01732-^
*4
=
=z,
•*6
=
J7
=24
s8
=
z3
s9
=
z6
•^10
=
J'a
J11=;P1
s12
=
^4
,,
=
s3
=
s5
dtc
dtc
=
P31
=
0.:
0.015-^
0.01732-^
z2
81
•*13
—
*15
=
-*17
=
J19
=
s2l
=
s23
=
^3
*u
=
ys
*16
=
r.
s\s
ia
J20
siny
s22
~
4
—
4
50
—
yb
Jh
—~r
—
h
z5
—
50 cosy
=
50 cosy
(zt... z6 and yx... y6 are given in table IV)
ST2
thyratron simulator replacing the effect of the
they are connected in a bridge circuit
=
7.2 Machine with variable
The
of variable
case
speed
multiplier
by solving the following
thyratrons when
speed
was
calculated later
on
became available. The sin y and
channel
six
two simultaneous
when the absent twocos
y
were
calculated
equations:
dz,
(108)
-JT=-vzi
dtc
P-f*i
(109)
dtc
giving the solution
Fig.
zx
=
cos
z2
=
sin y
that:
y
26 represents the
Since the
frequency
analog
scheme used to obtain the sin y and
is not constant in this case, two
(y z2
equation
•used in order to form the functions
obtained from the mechanical
•
and y
•
of the
multipliers
zt). The quantity
cos
y.
have to be
y was
synchronous machine,
being:
J.^+Q.mm+M.
=
at
where
(110)
Ma is the torque required by the load from the shaft of the
chronous machine in
82
M.
kg*m.
syn¬
Ma is positive in the case of the synchronous generator.
Ma is negative in the case of the synchronous motor.
Dividing both sides of equation (110) by the reference torque in order
per-unit form,
transform it into the
<»n-J-a>l d2y
co2
dy
dx2
Ps„
dx
Psn
we
to
get:
since,
substituting:
(O„-J-(0*
=
co2
Q-
=Qr,
—
mechanical time constant,
Trn
=
we
get:
dx
dx
For the machine under consideration, the mechanical time constant
was
T^
found to be 138.1, and
q
qr
=
0.00585
=
0.0416
J/rad/s
Thus the mechanical
=
0.0006
equation
kg*m/rad/s
becomes:
138.1^4 +0.0416^ +^; •*;-*;•£
dx
=
dx
K
Introducing the time scale given by equation (75),
5.524
d2y
+0.0083
-jf-4+0.0083—"
dtl
dy
-j-
dt.
+fi ird-i],rd irq
equation gives dy/dtc as a function
analog set-up is also shown in Fig. 26.
This
7.3 The
we
get:
(112)
Mra
M'a.
This
was
solved and the
thyratron simulator
Inserting
duces
of
=
(in)
the
thyratrons in the stator of
containing dead zones.
a current
the
synchronous
machine pro¬
The function of the
simulator is to determine the ignition and extinction
points
thyratron
of the currents
83
Analog computer
Fig. 26
scheme to generate sin y and
cos
y where y is
not constant
zt
=
50 sin y
z2
=
50
z3
=
o.oo5
z4
=
z5
=
cos
Px
P2
P3
y
oft/;-*;©
—18.1 V
=
=
0.3621
=
variable
=
0.0075
100% reference torque
-o.oo5#; $
^; /;
z6= -0.Q2
Fig. 27
diagram of the thyratron simulator for
synchronous machine
Block
controlled
S1...S11
My...MyS
(?!... G15
A
54,
A
55, A 57
A 56
=
Schmitt
=
bistable multivibrators
gates
=
integrators
=
inverters
!/„
Ux
=
ST\
—
voltage proportional
voltage proportional
=
is
angular frequency <orm
to the
firing angle
a
period corresponds to 2 Tc and whose
constant and independent of frequency
saw-tooth having the
delayed by
84
to the
saw-tooth whose
amplitude
ST2
three-phase
triggers
=
=
the
a
time
same
characteristics
equals Te
s
as
ST I but it is
Ml
S6
Gl
—
G2
^92
G3
-*g3
Ga
-*9a
G5
^-95
G6
"^96
G7
^~g7
G8
-*g8
g,
M2
U0..60
S7
M3
U«.120
S8
ST1
MA
U—iso
S9
M5
U..240
SlO
M6
U<x«300
Sll
M7
u«
®-*H
1
r
p_
^
S12
1<
U«.60
i
^
L
6'—*—^
M9
U«'I20
SlA
?l
5'
r
»
^^99
G10
-~9>0
G11
-—gn
G12
—*-9i2
G13
—^913
Gk
—*91A
G15
[—*-
M10
U«.180
D
6
*
^
Mil
U»»240
S16
"»
»
M12
U~>300
S17
*.
u360
M13
I,
©-
«>
5
->'
.^
^~
MM
,
**
-ll ®-
'1
»
M15
?
"
h
"5
®—
85
—*-
97
—,
1—
G8'
-*»98'
Gff
-*g9
1
G9
L
—S—
ST 2
S15
G7'
r
3
S13
I
9i5
f—
u»?
IOOpF
K
-| l«i k ft |
OA5l
1
-120 V
r—1100fcfl|
-fiookiTH- A55>
A5A
Fig.
28
variable
Circuit
generating
a
saw-tooth with
saw-tooth, having
the first
=
the second
saw-tooth, with the
cept that it
begins
transistors
defining
T3
=
zero
a
same
amplitude and
to 720°
characteristics
as
ST 1
ex¬
360° after it
the
points
of intersection of sin 7 with the
axis
amplifier
Schmitt
trigger
used
pulse shaper
7-4
=
a
T5
=
bistable multivibrator
T6
=
monostable multivibrator
Ti
=
a
T8
=
amplitude comparator which gives
triode used
STl
=
as a
as a
switch which ends STl
a
sudden
change
when
U360
Tg
=
monostable multivibrator
T-io
=
a
Ua
=
^360
=
86
constant
period corresponding
=
STl
T»T2
a
frequency
STl
=
jiookciH-
A56
ST1
triode used
as a
switch which ends STl
voltage proportional to the circular frequency a>rm
voltage, which equals half the amplitude of any
STl and which corresponds to the angle 360°
a
a
of STl
or
flowing in each phase and
the
inputs
of the
to build the necessary
integrators
A
28,
gates required
A 34 and A 42 of Fig. 25. The
to drive
following
describes the apparatus built to simulate the thyratron effect in
our
problem.
7.3.1 Block
diagram
Fig. 27 shows
block
a
the
diagram for
thyratron simulator.
comprises
disposed 360° from the
two "saw-tooth
generators",
each of which is
other and each is
synchronized
with sin y. As will be
of this saw-tooth is constant and
frequency,
its
period
should
independent
correspond
of the
given,
amplitude
the
For each
frequency.
electrical.
to 720°
which determine the
It
begin
The six
ignition points
negative
parts of each of the three stator currents
of the
were
positive
obtained
by
and
com¬
paring each saw-tooth with six voltages displaced 60° electrical (Ua-,
£4'+60---£4' + 3oo)defines the
The
beginning
firing angle
a! between the
pulse No.l—which
phase No. 1—and the func¬
adding a d.c. voltage (proportional to the
voltages. The six points defining the ends of
of the stator current in
tion sin y could be varied by
angle a')
the
to each of the six
positive
and
also obtained
negative parts of
by comparing
For every gate, it
was
required
follower and the
each of the three stator currents
them with the
to build a bistable
transistors. The
gating
trigger the corresponding
made to
According
to
table IV,
we
these gates
are
every 2
similar in their
are
Fig.
27
one
timing,
are
given
were
axis.
multivibrator,
pulses defining
a
cathode
each gate
were
bistable multivibrator.
need 18 gates, and since the
different from
multivibrators. In
zero
another,
but
then
need in fact
we
as
timing of
for the
12 of
remaining
only
6
15 bistable
the 15 bistable multivibrators and the
18 gates.
7.3.2 The "saw-tooth" generators
The solution of the mechanical
equation (112) leads to the achievement
quantity y. Benefit was made of this fact
in order to generate a saw-tooth that is synchronized with the function
sin y,and that has a fixed amplitude irrespective of the circular frequency y.
This quantity y was given into the input of an integrator. Supposing that
the integrator is stopped from integrating after a period of Tc seconds,
of
a
voltage proportional
to the
87
integrating condenser is discharged, and the saw-tooth is allowed to
begin another period, then the maximum value achieved by the saw-tooth is
the
equal
to
dtc
\ y
=
2
n
o
Therefore
have to expect that the maximum value is constant and
we
frequency. The maximum value will in this case corre¬
independent
the
to
angle 360° electrical, we shall denote it by U360. Since we
spond
to
are going
compare this saw-tooth with voltages corresponding to the
ignition points, i.e. with Ux., U0l,+6O...Uai,+3OO, then we cannot change
the firing angle a' outside the range 0°... 60°. In order to extend the range
of the
tooth
allowed to exist for
of
a.',
Tc
and the maximum value in this
the
saw
electrical. So it
was
possible
was
tween 0°...360°. But
to
change
had to
we
after 360° electrical from the
a
period of 2TC instead
will correspond
case
the
firing angle
angle
to the
of
720°
a! in the range be¬
generate another saw-tooth that begins
beginning
of the first one. Then, again by
comparing this saw-tooth with the six voltages, we got the six ignition
pulses which appear 360° later than the first group of pulses. Thus we
obtained the train of ignition pulses from the first saw-tooth for the first
360°,
and from the second saw-tooth for the second 360° and
two trains of
pulses
then added
were
Integrator
A 54 of
integrated,
and after 720°
Fig.
get
a
These
by and-circuits.
27 allows the
we
so on.
voltage corresponding
pulse
that
discharges
the
to y to
be
integrating
condenser.
Considering the Fig. 28, tube T-,
in
a
time of 500
us
which
can
acts
as a
switch
neglected
be
discharging the condenser
when
compared with
at the reted
equals
frequency.
period
T2 determine the points of intersection of the sine wave with the
that
tooth
2.5132
Then the bistable multivibrator
pulses
every 720°. These
s
consisting
pulses
were
the
saw
Transistors
of the double triode
zero
7\,
axis.
Ts yields
widened in the monostable multi¬
produced pulse was
discharge
integrating condenser.
The second "saw-tooth generator" functions on the same principle, but
instead of the extinction pulse being obtained by comparing the sine wave
with the zero axis, it was obtained by comparing the first saw-tooth with
the voltage U360. This was done by the comparator consisting of the
vibrator
composed of the double triode T6.
made to control the triode T-, used
double triode
T8.
The rest of the circuit functions
exactly
controls the first saw-tooth generator.
88
The thus
the
to
as
the
corresponding part
that
7.3.3 The
The six
coinciding voltages
voltages which—-when compared with the
tors—produce
in
ignition pulses were
of six d.c. amplifiers,
the six
Fig. 29. It consists
voltage
corresponding to the
sponds
to the
angle
angle a'(t4-)>
60 + a'((/6O+(r.) and
saw
tooth in compara¬
by
the scheme shown
formed
the output of the first is
*ne
output of the second
so on.
Thus
we see
that,
a
d.c.
corre¬
for any
angle a', the difference between each two successive voltages corresponds
to the angle 60°. The angle a' was varied simply by shifting these voltages
in the vertical direction which took place by varying Ux. that is common
to the input of the six amplifiers.
The sudden variation of the voltage Ua. results in a sudden shift of the
six coinciding voltages. Thus the apparatus is effective not only for slow
variations of a', but also for sudden variations. It responds also quickly
to any sudden
change in the
7.3.4 The Schmitt
The Schmitt
It has two
trigger
trigger used
to
the saw-tooth and the coinciding voltage. When
equals the coinciding voltage, the Schmitt trigger
changes its condition, and
tage which,
~Ua-,
ages
we
obtain
a
differentiated, yields
when
Scheme
—
produce the ignition pulses is shown in Fig. 30.
inputs, namely
the first saw-tooth ST I
Fig. 29
value of y.
producing
£/„' + 60>
proportional
—
the six
a
change
in the output vol¬
negative pulse.
coinciding voltages
^5t' + 300 are VOlt^oc' + 240>
^t'+180>
angles <x'°, (a'+ 60)°, (a'+ 120)°, (a' +180)°,
£4'+120>
to the
sudden
—
—
—
(a' + 240)°, (a' + 300)° respectively
^lso
=
U'
=
voltage proportional
to the
angle
180°
-100 V
89
The second saw-tooth ST2
STl in addition to the
another Schmitt
These two
sisting
negative pulses
added
were
beginning
given to
we
got
a
together by the and-circuit
of the two diodes OA5 and the transistor OC71.
triggers
7.3.5 The
positive
and
Fig. 31
the
end-points
serves
negative
triggers,
of the currents
to determine the end
one
part of the current, and the other
of the current with the
points of each
Effectively, it
of which is sensitive to the
to the
Two p-n-p transistors form the Schmitt
the other Schmitt
Fig. 30 shows
parts of the stator currents.
consists of two Schmitt
portion
con¬
and the and-circuit.
pulses determining
The circuit shown in
of STl. This
the input of
negative pulse.
were
trigger, and when they coincided,
the two Schmitt
of the
360° after the
comes
coinciding voltage
positive
negative part.
trigger
that selects the
positive
axis, and three n-p-n transistors form
zero
trigger.
7.3.6 The bistable multivibrator
This bistable multivibrator
—24
V, and
was
was
fed from two
used to obtain
—22 V and +22 V. The
cause
a
voltage
rectangular
for
choosing
wave
such
sources, +24 V and
that varies between
high voltages
for the
given
begin and end of each gate as given in
table IV—were made to trigger the bistable multivibrator. Fig. 32 shows
the bistable multivibrator triggered by the pulses 1, 1', 4, 4'.
bistable multivibrator will be
The
pulses—determining
in the next section.
the
7.3.7 The gate
anticipated from the gating circuit is to allow the volt¬
(constituting the derivative of the stator current) to be in¬
age given
tegrated during a pre-determined period of time, and to earth that input
outside that period. Ideally, the impedance of that gate should be infinite
during the integrating period, and zero during the nonintegrating period.
The function to be
to it
single phase investigation, an error has arisen
impedance of the gate was not zero during the nonand
there was a voltage drop across the gate during
integrating period,
We have
seen
that in the
from the fact that the
that
period. This voltage drop was of the order of 0.3 V, which was inte¬
grated during the nonintegrating period.
In case that the integrator multiplies by a factor more than unity and if
the ratio of the
error can
90
nonintegrating period
be troublesome.
to the
integrating
one
is
big,
the
ST24
ST1 6 6 -Ue,
Fig. 30
Schmitt trigger producing
one
of the
ignition pulses (pulse No. 1)
+6V
5
m
-6V
rx
I
6'
O.OIuF
x
&ti
%
m
50pF"
—|15kfth
OCV.0
0C71
0C71
i
120 knt
50pF
~r
y
"
OCV.0
OC1A0
c
o
S
I
Fj'g. 3/
Schmitt trigger determining the end
=
current in the stator
3
=
pulse produced
6
=
a
it
rectangular
winding
No. 1 of the
at the end of the
wave
defining
points
point
of the current
i
it
synchronous machine
positive part
the end
1
of the current il
of the negative part of the
current i.
91
t^H-
Leer
-
Vide
-
Empty
,
1
0.1tuF
A
f
i'6
OC77
-M,
x
to
I
#
I
OC77
X
m
T
driving the integrators ,428,
I
^QtuFJL
0A5
One of the gates
JL
0A5
that controls this gate
Fig. 32
+24
OC77
-24V
4* A'6
OC77
i
OC77
'
I
2N1304
g7
A 34, ,442 of Fig. 25 and the bistable multivibrator
JL
20A5T
n
OA5$
n
—
This is indeed the
case
in
two or more, and where the
the
multiply
problem
our
where this ratio may be
integrators
integrated input by
A
parallel branches,
diode), only a voltage drop
each
the rest
and
positive
composed
of 0.02 V
of
negative voltages
(which
consists of
transistor in series with
a
the
was across
the diode. The diodes
was across
to
factors which may reach 50.
It has been found that in the above mentioned gate
two
equal
28, A 34 and A 42 of Fig. 25
were
a
transistor, and that
necessary to isolate the
from the collectors of the p-n-p and the
respectively. This isolation was necessary in that case,
voltage of the preceeding multivibrator which steers the gate
n-p-n transistor
since the
ranged
between + 3 and
the p-n-p reached
—
value
a
3
V, and when the voltage
more
positive than
+3
at the collector of
V, the diode consisting
of the collector and the base would conduct.
voltage
Hence the collector
4- 3
V, if the diodes
ages, and the
were
voltage
would have been
absent. The
across
clipped
at the
is also valid for
same
the gate would have
voltage of
negative volt¬
ranged only between
+ 3 and -3 V.
If, however, the voltage of the preceeding multivibrator
range between
voltage
across
+22 and —22 V, it would have been
was
made to
possible
for the
the gate to take any value between +20 and —20 V with¬
isolating diodes, and without
clipped. But we should search
out the need of the
any
voltage would be
for n-p-n and p-n-p
transistors whose base-emitter
respectively.
and +20 V
junction
can
second type. The last part of
(a)
Advantages of
the
It works for variable
cessive
voltages
of
—
20
(Philips) is a good example
(Texas Instruments) for the
The transistor OC77
for the first type and the transistor 2N 1304
7.3.8
withstand
danger that that
Fig.
32 represents the gate thus formed.
thyratron simulator
frequencies,
the
ignition pulses being always
angle
between every two
60° for every
suc¬
frequency.
firing angle a' can be varied between 0° and 360°.
(c)
voltage drop across the gate is 0.02 V when it is closed, for input
voltages not exceeding +20 V.
(d) It is effective when the variation of the firing angle is slowly or sud¬
(b)
The
The
denly executed.
denly varied.
94
Also when the circular
frequency
y
is
slowly
or
sud¬
8. Analog
investigation
/,J,C. flowing from the positive pole of the d.c. power supply
smoothing inductance is equal to the sum of the currents in
the three thyratrons I, III, IV of Fig. 4. It equals also the sum of the
positive portions of the currents i[, ir2, ijj.
The positive parts of these currents are selected and added by means of
the circuit shown in Fig. 33 where the diodes perform the job of selection.
An oscillogram for the current i^c_ is shown in Fig. 34, which consists
The current
the
through
is to be
expected of a d.c. component and a pulsating part with a funda¬
frequency equals to six times the fundamental frequency of the
stator current. In this figure are also given oscillograms for the stator
current and phase voltage.
as
mental
The noises encountered in the differentiation of the fluxes to obtain the
phase voltages are noted in the voltage wave. These noises are to be found
only in the voltage wave, but the currents are free from these noises since
they are obtained from integrators. This differentiation was avoided when
the system
was
in
9.
chapter
simulated
oscillogram for
Again as in the case
An
on
the PACE
the instantaneous
of the current
analog computer
wave
as
will be
of the torque is also
/j_c, the torque
wave
consists of
given
given.
a
d.c.
400 ka
0A5
1
100k£l
-Oh-1=
o
2
0A5
100 ka
OA5
100 kn
O
3
O
Fig. 33
Analog
1
=
0.25
2
=
0.25
i[
/;
scheme to obtain the instantaneous current
3
=
0.25
4
=
/d'.c.
/,d.c.
ir3
95
component plus
pulsating
a
one
times that
of the stator current
The
is
the
same
same
ird
Fn>
and
34
V,
=
I
flowing
irq
If
in
fundamental
i]/j and
Urf
Phase
in
the
of the
=
cunent
Lrd i\
2
same
io'm
100%
/' and the
=
=
I
current
id
anient
/,;
01/;;
I
^rtf-rnti
96
0 25
Phase
Lrdi\
cunent
2
i\
=
and
a
20V
=
which
are
also gi\en
Iq and also the
three-phase system
l" H F±t-
1
frequency equals
figure
..TTT
20V
/"/? J4/>
i//J
damper windings ir0
the
shown
are
16%
0 25
=
also valid for the fluxes
Oscillograms
F/? 34a
a
six
in
figure
The currents
rents
with
and the
-0
25/,;
=
115
cur¬
20V
rjh.
\i\i^flpMtf\fifa'
Fig. 34c
1
Phase current
L', t'
0.25
=
i[
and the
damper winding irD
the
current in
WW
X.
2
/
2oy
Tl
t
-'tll.iJ
2oy
i
n
/•/,£,' J-/</
i-ifVtrfcftl
Phase
i=o.25l;«',
current
i[
2=
20V
2
and the
current m
the
-/J.
^
L
p'
I
,_•
I
v/'
^
V
\n\m
-+t
in; .?•/('
=
V-
,.
r-~\
20V
I
dampei winding i'Q
0.25
Phase
L'di\
cuiient
2
i\
=
and the field current
\'s
0.2/rf
97
1
v/AAAf
Phase
Fig. 341
1
=
0.25
'[£\\
Lrti\
i\
cm rent
2
and the
A/;
-0.05
=
clectiomagnetic torque \fre
20V
20V
i
=0.25
/J
Phase current and the d.c. current
/•"(?. 34g
0',
2
j I
t_
025/.;;^
=
"i
r
]
i
i
i
I
'
'
I
1
i
r
i»i
20V
5V
//,!,'. .?•///
1
Phase current
=0.25/^1
98
2
;J
=
and the
0.075 u\
phase voltage u[
i
!
I H
'
i"
'T
20V
'
".
'
'"
'-Mrm-Hi
—
-
/
^ n
t
Hp-h
<
v
r
n+]iTit4fHt#
In;
1
?•//
=0 25
Phase
Lji\
current
2
i\
=
and the flu\
i/, linking
the i_oil </
-Oli//,",
20 V
ijjd
/1i3\tji
,20V
rf
//!,' 34/
I
Phase ument
=U25Lrjt\
2
i\
=
and the flux
ijir linking
the mil c/
2!//;
99
9. Simulation
of the system on the
PACE
analog computer
9.1 The mathematical formulation
As
was
shown in
chapter 8, the mathematical formulation
of the three-
multipliers when y was assumed to be
constant, whereas in the case of variable speed, 12 multipliers were needed.
As generally accepted, differentiation is to be avoided when possible. Be¬
cause of the limited number of available multipliers, we were obliged to
phase system required
carry it
on.
A
use
of 8
larger computer
thus it became
cally in such
the
a
possible
was
then established in the
to formulate the
way that
no
institute, and
three-phase system mathemati¬
differentiation is necessary. The
following
simulation was made in co-operation with Mr. M. Mansour, who is mak¬
ing
other
investigations
on
this type of machine.
Fig. 35 shows the connection diagram of the three-phase system, where
each of the d.c. supply voltage Ud and the smoothing inductance Ld is
divided into two halves.
Now for
phase
No.
1, when
V
(i) thyratron
No. 1 is
conducting,
we
have
u'{
=
(ii) thyratron
No. 4 is
conducting,
we
have
Hi'
=
ur
'W
(iii)
both
thyratrons No.
1 and No. 4
are
—\-£
nonconducting,
we
ur
(113)
have
«i=-^-+"o
ax
For
phase
No.
2, when
(i) thyratron
No. 3 is
conducting,
(ii) thyratron
No. 6 is
conducting,
(iii) both thyratrons No. 3 and
we
we
No. 6
u'{
=
have
u'Z
=
are
urL
Ud
have
—
[Urd
I
(114)
nonconducting,
ax
100
urL
we
have
05 U
w
157
is
05 IL
HJ
II
05 LU
:ro
05 u.
Ml,
F/g. 35
v
v
Connection diagram of the stator of the
synchronous
phase
For
v
^.
No.
No. 5 is
conducting,
we
have
u'3r
(ii) thyratron
No. 2 is
conducting,
we
have
u'3r
both
thyratrons
No. 5 and No. 2
are
Urd
urL
=
=
—
I
(115)
dV3
=
dx
=
—
nonconducting,
u,
ur0
controlled
3, when
(i) thyratron
(iii)
three-phase
machine
we
have
+W
0. Since the machine is star connected.
"0
'd.c.
=3("ir+"ir+"3r)
=
lvl + lv3 + 'u5
=
(116)
'l++'2+ + J3 +
fas
uV
The
ud
'd.c.
=l:
(117)
dx
voltage equations
of the
synchronous machine
are
-R'.il+lp-y.p
dx
dird
dirD
dirf
--Z + -l-Hxq- iq
+
R'.id + xd--± +
v
dx
dx
dx
„.
=
T
irQ)
101
o
=
d \brn
»rD+n-p
=
dx
«;
=
d i'j
d irf
d irn
i'D+rD-xd(i-odD)-^+rD(i-nD)-J.+T,D—Ddx
dx
dx
d \Jjrr
d fr
d irn
d irr
dx
dx
dx
dx
R'-/;+^+y-^
dx
=
R'.i'q + Xq^
dx
+
-^+y(xd-ii + irD+irf)
dx
d\l/r0
dira
,
.
diT0
dx
-i[r]
=
dx
[Lr]_1
[r]-1K]-[r]-1.[RT.[r]-y[r]-1[L"].[r]
(lis)
representing
the different
was
calculated and the numerical values
substituted in the matrix equation
constants of the machine were
in the
resulting
—
=
dx
—
=
set of
13.45— -0.01934
dx
di'f
([/}-/})+0.2028 fD
-5.772— -0.06005
d\l/d
=
dx
-1.037—
dx
(118)
equations:
dx
dx
—*
following
([/}-/})-0.2307 irD
+0.07128(1/}-/})+0.1130 irD
di'
dijj'
—q- =13.19-^+0.2710/^
dx
dx
d in
—^
=
dx
d \bra
-2.540-^ -0.07273
dx
irQ
where,
-P-
=
dx
and
102
-p
=
urd+y (0.2684 Lp+/rQ)-0.0571 i\
ur-y(0.5805 ira+?D + ?f)-0.0571
(119)
ir
(120)
The transformation from the
drd
dirq
dirx
dt;
dx
dx
dx
dx
is done
to the
according
dirq
did
dll
dx
di"d
dip
—JL
.
cosy--—
dx
—-=—-
dx
equations:
.,
.
smy-y
dir
.
-smy + —--cosy+y
dx
dx
=
—
dx
.r
(122)
ix
•
to obtain the derivatives of the currents
possible
Out of these it is
(121)
I,
i[,
'3-
2>
di\ _dira
==
77
dx
.dX
dir2
-0.5
=
~dx"'
dx
dx'
£
dx
Thus after
dV,
dx
dx
2
of the
introducing the conditions
i[, ir2, ir3.
thyratron action,
we
obtain
the stator currents
But
are
we
need to build
made
according
d\li'
dVi
and
—-.
dx
The simulation of these
dx
to the
quantities
equations (119), (120) and thus no differenti¬
quantities urd, urq, we need the
ation is needed. In order to obtain the
quantities u\r, u'2T, u'3r
Urd,
urL
was
djj\
uL,
——,
dx
simulated
—^- is equal
is
di>r2 d$\
——,
——,
dx
are
u'0r
dx
according
to the
dx
whose constituents
part of
to the
—-
dx
equation (117),
during
the
period
where
in which thyratron No. 1
conducting.
dVa &Vp
—-
dx
,
d^
——=
dx
are
.tAi
related
to
dx
dx
dvt
—~
dx
dVa d*'
cosy
dvq
-
——
dx
,
dx
by
.
the fellowing
equations
.
smy
-y
typ
103
=
——
dx
siny+ —-2-cosy + y-^a
——
dx
ax
This simulation is, however, to be avoided
with
loop
a
total
gain equal
to
unity.
The
it results in
as
reason
why
an
an
algebraic
algebraic loop
has tobeavoided is that it tends
This
toamplifyany noise present in the system.
algebraic loopwasavoidedbycarryingon the following substitution:
Vd
=Vd + x,,- <JdD ird + fiD if
•
diprD
dVd
-^-
=
dx
dird
dx
dx
dVq_dj^^
~
dx
dx
Thus
—
-
siny
+Xq<Tq'
dx
=
—cos
dx
dx
TD
dird
y
^__A^
dA
dx~
T'Q+Xq(Tq' dx
-
[(xdadD-Xqaq)
-
(123)
-J
+
+ sin7
(^D
-
dir„
Substituting
the numerical values of the machine constants in
(123), (124),
we
=
siny (0.0001
\
—+ 0.0205
dx
ir0)
J
+
dx
dVi
dx
-cos7
+
siny
(
\
irB)
dil
0.0001 —^ +0.0205
dx
\
ir„
(0.0206 -^ -0.0177 irD]
+0.0759
41 +7(^-0.0759 Q
dx
104
equations
cosy( 0.0206— -0.0177 irD
dr.
=
-
get:
+ 0.0759—--y(il/rB-0.0759
-^
-
(124)
dx
dx
-
y(.^p-xd(TdDire)
+ xitriD-f£-+HV.-xiaiDQ
—
dirf
+nD--—
dx
dx
—
+
di\
dV,
irD
-=r+xd-<rdD
\(xd<rdD-Xqcq)- -j +cos^D-
+ xdadD~,
—-
dirf
-j—+xd-(TdD-—+nD--—=
dx
Thus, the gain of the algebraic loop is effectively reduced
stead of
to 0.0206 in¬
unity.
d\l/\ d\\ir2 dfc
The derivatives
dx
,
formed according to
were
,
dx
dx
dx
dx
dx
dx
2
dx
dx
dx
2
dx
quantity u'0r was simulated according to equation (116).
thyratron effect has to be taken into consideration in the simulation
of the voltages u[r, u'2r, u'3r and we have to follow equations (113)... (115).
Ideally, these conditions put on these voltages should be enough to give
The
The
the dead
in the stator currents, i.e. the derivative of each of the
zones
stator current will be zero
selves have to be
But any minute
during
wir>
period,
thus
are not
right
of the currents
i\
=
-r
—
are
made at first in the fixed
=
0,
tu
then
=
'«
<
iB
0,
=
—p,
i.e.
when
di*
—
i[
voltages
difficulty,
we
the interval
0, then
i.
=
—=,
—
•
=
was
co-ordinates
made
on
the
men¬
the derivatives
0
=
dx
i.e.
—-
dx
di*
1
diP
i.e.—-
has to be isolated from
=
axes
(1, 2, 3) directions, then
dx
0
dx
that the
i\, if, accordingly
^3
Thus
so
being integrated during
isolating the derivatives
0, then /;
r,
i2
ir3
small current to flow
In order to avoid this
then transformed into the
tioned condition of
when
a
the comparators
more.
any
causes
current has to flow.
no
(a, P), and
i
when
activating
prevent the derivative from
Since all the calculations
when
the intervals where the currents them¬
in the derivative
u'i
have to
where
error
that
ui>
during
zero.
—=
J2,
=
d*
=
y3
—
dx
being integrated during
the intervals
0.
105
is
to be taken from
dirB
is
Ji
to be taken from
dx
dig
—
is to be taken from
dx
As for the time
time
on
when
—=.
dx
—-
di'
1
dig
—
dx
(
1
di'x\
-=
\ ^3 dx)
i2
=
,
when i3
The
was
taken to be
unity, i.e.
2nf„-t
following gives the scaled equations:
for phase
No. 1
1 conducts
(i) thyratron No.
1
8
(ii) thyratron
0.25
0.25
>'u'<
<=
u'{
8
No. 4 conducts
(iii) thyratrons
=
*
No. 1 and 4 not
—(0.25
dx.
conducting
iAr,) + 0.25 u'0r
for phase No. 2
(i) thyratron
0.25^
2
(ii) thyratron
No. 3 conducts
=
^-^
8
8
No. 6 conducts
'
0.25«"
'
(iii) thyratrons No. 3 and 6
0.25
106
u'{
=
0
not
conducting
—(0.251/4)+0.25 u'0r
dt.
x
=
il
and
scaling, the relation between the per-unit time
the computer tc
=
„
=
equation (122) when both rj ^ 0,
that the relation between t and tc is
tc
0
x
^ 0.
and the
tc which
means
for phase No. 3
(i) thyratron
0.25
u'3r
No. 5 conducts
=
3
«-(£-*
u'3r
No. 5 and 2 not
=
d
—(0.25
dtc
conducting
ij,r3) +0.25 u'0r
0.25u£ =-(0.25 m';+ 0.25 u2r +0.25 M'3r)
0.25
ul
=
rf
—(0.05
dtc
5LJ
ird)
=
[A (0.05 rpl)+ -^-(0.05 i*3) -^-(0.05 ,*,)]
+
d
2.69—(0.25
dtc
+ 0.0507
d
-—(0.2
d
tc
irD)
=
d
-4.618 —(0.25
dtc
+ 0.0480
d
—-(0.25
dtc
=
-1.037
—
dtc
(0.5
—(0.5f£,)
fQ)
=
irf)
-
0.2307
(0.2 irD)
==-
"
—(0.25 #J)
VrA
(0.25 i})+0.1413 (0.2 Q
4-
—
rfr
=
(0.25ipd)+0.1782(0.1
2.638—(0.25 </r,r)+0.0271 (0.5 irQ)
5.080
—(0.05 Q
dtc
(0.25
<^)-0.1201 (0.1 Urf)
d
i})
-0.0713
—
i/^)-0.0097 (0.1 urf)+0.0039 (0.25 irf)
(0.2 irD)
5.080
and
8
No. 2 conducts
(iii) thyratrons
0.25
—
8
(ii) thyratron
025
—-
(0.25 ^)
-
0.0727 (0.5
irQ)
(0.25MJ) + 0.5y(0.5^)-0.1428(0.1 ird)
Analog scheme for
Fig. 36
the simulation of the
three-phase controlled
synchronous machine
Sl
=-0.05^
54
=0.005—'-
s7
=0.05
dx'.
s2
=
s5
=
s8
=
dt,
dt,
dtr
dtr
0.25
"
A
-0.25
dM
s13
=
s15
=
25
=
25 (0.0001
s14
dtr
=0.05—5
s9
=-l(U'd-u'L)
=
—
dt,
d\\
Sl2
dtr
atr
=
0.25^
a
tr
(0.5^-0.038^)
( -0.0206 -^ +0.0177 i'D
*
dt.
di'
(
\
0.25
—
dt.
+ 0.0205
diK
F,
-0.25
dt.
s20
s6
dt,
(t/;-«D
Si
=
.dil
-0.05
di d.c.
-0.05-
dirx
0.05
s3
dt.
di\
—
-0.05—2
0.25
dt„
dt.
(0.5 ^-0.038/;)
=
P3
=0.0577
=0.0577
P6
=0.5000
0.8660
P9
=0.3795
Pl2
=
0.25Urd
0.1000
P15
=
0.1000
0.1000
P18
=
0.1000
P20
=
0.1428
P21
=
0.1428
P23
=
0.1000
P24
=
0.1000
0.3253
P26
=
0.3325
P27
=
0.3325
=
0.3577
P29
=
0.5773
P30
=
0.0507
P31
=
0.0039
P32
=
0.0271
P33
=
0.0480
P34
=
0.2307
P35
=
0.4618
P36
=
0.1413
=
0.0713
P38
=
0.1037
P39
=
0.2500
PI
=0.2690
P2
=0.2638
P4
=0.1000
P5
P7
=0.8660
P8
=
P10
=
0.5y0
Pll
=
P13
=
0.5000
P14
=
P16
=
0.1000
P17
=
P19
=
0.1000
P22
=
0.1000
P25
=
P28
P37
108
1.25
L'd
P40
=
0.2903
P41
=
0.5080
P42
=
0.0727
P61
=
0.005 <x'°
P62
=
0.5000
P63
=
0.8660
P43
=
0.1342
P44
=
0.2136
P45
=
0.1468
P64
=
0.0020
P65
=
0.0019
P66
=
0.0007
P46
=
0.1922
P47
=
0.0550
P48
=
0.1025
P67
=
0.0011
P68
=
0.0009
P69
=
0.0008
P49
=
0.0100
P50
=
0.0100
P51
=
0.3795
P70
=
0.0724
P71
=
0.1
P72
0.0097
P52
=
0.5000
P53
=
0.8660
P54
=
0.8660
P73
=
0.1782
PI4
=
U}
U}
=
0.25
P75
=
0.1
P55
=
0.3795
P56
=
0.3795
P57
=
0.0724
P76
=
0.1201
pii
=
o.i
[/;
P78
=
0.3671
P58
=
0.25 Mrn
P59
=
0.0145
P60
=
0.0003
P79
=
0.5
109
U}
U}
Leer
-
Vide
-
Empty
dtc
(0.25 Vq)
=
(0.25u')-0.5y(0.5^)-0.1428(0.1 irq)
j
A
-—(0.05 Q
a
tc
—
a
=
fc
(0.05 Q
=
cosy
A
-—(0.05 Q
-
atc
-—(0.05
sin y
atc
i,')-0.5y(0.1 ij)
siny •—(0.05 ©+cosy -—(0.05 lrq)+0.5y(0.1 ©
a
fc
a
fc
-£-(0.05 1^)= —(0.05 Q
dtc
dtc
-£-(0.05 f2)
=
-£-(0.05 r3)
=
a/c
—
-0.5-£-(0.05 0+ ^ -£-(0.05#
•
2
dfe
af,.
-0.5-£-(0.05O-^~(0.05,-J)
2
afc
(0.25 <K)=(0.01). sin y
(0.01) -cosy
-
+
—
dtc
(0.25 ^J)
=
+
-
+
—(0.25^)
=
atc
0.1468
[0.055—(0.05 i J)+1.025 (0.5 fc)
[-2.136—(0.25 ^)+0.3671 (0.1 Urf)
(0.25 i}) -1.922 (0.2 Q
0.3795—(0.05 O
-
atc
(0.01)
•
cos
y
-
0.5 y
|o.055
l_
—
dfe
[0.5^-0.3795(0.1 /J)]
(0.05 g+1.025 (0.5
jre)l
inyf -2.136—(0.25^)+0.3671 (0.1 Urf)
(0.01)-sin
»]
0.1468(0.25 i})-1.922(0.2i
0.3795—(0.05 $+0.5 j>[0.5 ^-0.3795(0.1 Q]
-£-(0.25 ft)
111
-—(0.25^5)
dtc
-0.5
=
(0.25 #5+0.8660— (0.25 M)
—
dtc
—(0.25^)
=
—(0.5?)
=
-0.5
dtc
(0.25 «/#-0.8660—(0.251/^)
—
dtc
dtc
dtc
f [0.0145(0.25M^)+ 0.0724(0.5^)(0.1 Q
J
dtc
-
(0.5 ^r) (0.1 Q
0.0724
-
0.0003
(0.5 y)]
d tc
Fig. 36 shows the complete analog scheme, where the comparators
Mx... M9 take the job of replacing the switching action of the thyratrons.
In addition, 22 multipliers, one servo resolver, 79 amplifiers as integrators,
summators, inverters and high gain amplifiers (14 of them
and 79 potentiometers
In order to
to the
iT2
to make the current
begin
at the same time.
for the current
as
the first
action, the pulses No. 2 and 5
that when either
so
il ready
to
pulse No.
begin,
it
pulse
appears
was
sure
that this condition is also
\j/rq)
pulse
they
appears,
parator M10.
positive
and
negative.
9.2
were
are
only
switched
by
causes
represent the
by
on.
means
means
were
added
No. 5 appears
also the current
no
pulse
exactly at
the
This is done
a
closed
path
load
case
from
appears. In order
by
point when the first
means
of the
com¬
of a step function which is
at the moment of the first
This is done
or
realized, the values d/dtc (0.25 \prd)
made open and
It is controlled
integrators)
solved.
the moment of begin of computation until the first
to make
2
Thus, the problem of finding
It is also necessary that the system should
and d/dtc (0.25
are
needed for this simulation.
the inverter
No. 3 and 6
pulses
tending
to
begin
were
of the
normally
pulse it suddenly becomes
amplifiers 0441, A79)
of
Fig. 36.
Oscillograms
Oscillograms for the quantities of interest were taken for the case when
a.'
0. These are given
\, M'a
0.4, y0
170°, £/;
1.3, V}
1, y0
in Fig. 37i...37iv.
Fig. 37i shows the 3 stator currents (i[, ir2, ir3), the phase voltage u[, the
voltage of the star point with respect to the middle point of the d.c. source
=
(«o0»
tne currents in the
transient behaviour
figure.
112
=
=
as
=
damper windings
well
The current in the
=
the
steady
damper winding
as
=
and the field current. The
state are to be seen
possesses
a
on
that
d.c. component
which dies out in the
behaviour.
diTd di' di\
—
—, —-,
dtc
dtc
dic
steady
state. Also the field current shows a
ira, iTft i'd, i'q,
37ii shows the currents
Fig.
,
,
,
.
,
voltage across
and the
Fig. 37iii shows the transformed
similar
the derivatives
.
the inductance.
stator
voltages
in the fixed
axes
co¬
(urd, tfq), the
moving axes
voltage across the thyratron No. 1 (ttrvi), the voltage of the star point
with respect to the cathodes of the thyratrons 2, 4, 6 (u0rk), and the de¬
(ifx, Up),
ordinates
rivatives
—-
—-,
d
.
dtc
tc
dfc dty
tives
Fig. 37iv shows the produced electric torque Mre, the
\j/rd, ty\ linking
fluxes
,
dtc
,
windings d,
the
and the
—-
dtc
co-ordinates
those in the
speed
q, the fluxes
(i/>a, \J/rfi),
the deriva-
y.
oscillograms the behaviour of the transient component as well
steady state component from the moment of switching on of the
In all the
as
the
voltage Ud
are to
Fig.
37i...37iv in the
amplitudes
are
details of the
Fig.
be shown.
38i...38iv show
37
exactly
the
same
steady state. They are
amplified
oscillograms
Oscillograms
as
high
can
be
as
oscillograms given
for the different
seen.
quantities corresponding
three-phase
analog computer
=
1.3
Mra
=
0.4
IT,
to
=
a'
1
=1
Fig.
the recorder allows. Thus all the
easily
controlled synchronous machine
Ui
in the
extended in the time axis and their
as
=
to the
taken from the PACE
170°
Vo.= 0
Fig. 37i
(a)
=
0.1
i\
(d)
=
0.25
u\
(g)
(b)
=
0.1
i\
(e)
=
0.75
u'0r
(h)
(c)
=
0.1
x\
(f)
=
2
=
=
5
irQ
0.25
i}
rD
113
Fig. 37 ii
£/}=!
a'
=1
to
Urd
=
1.3
MTa
=
0.4
(a)
=
0.1
Vx
(d)
(b)
=
0.1
%
(e)=0.05-^
(c)
=
0.5
/;
(/)=0.05-i
dtc
Vo
=
0.5
\\
di'd
(g)
=
170°
=0
=
0.05^
w-ib-*
dK
Fig. 37iii
Urd
=
1.3
U}=1
a'
Mrtt
=
0.4
y0
=1
Vo
(a)
=
0.25
urx
(d)
(b)
=
0.25
urf
(e)=0.25u'Dl
(c)
=
0.25
urd
(f)
=
=
0.25
0.25
«;
=
170°
=0
(g)=0.25^
(h)
=
a'
=
0.25^
dtc
u'0\
Fig. 37 iv
Vd
=
1.3
V}=1
Mra
=
0.4
70
(a)
=
0.25
(b)
(c)
114
=
=
Mre
0.5
ft
o.5
yq
(d)
(e)
=1
=
=
Vo
170°
=0
o.5
r.
(g)=0.25^
0.5
0J
(h)
dtc
=
0.5 y
-
/
wvwwwwa/^^^
-
-
V
n
/.->
V
i\
n
n.
n
>n •''"•/
V
n
-t-
,
I,;
n
i in.;
n
n
^/vWv V
V V V V V V
vfiV
^Wy
v
V/
-
/-1 /• :/• /- /- /• /• /• /• /• /• /• /• /•/// /• /• /• /- /• /• /• /• /• / /•///// / /
y v v v v \ v v v www v v w \- w w w \ \ v \, \,
y
i)
n
/Ww/V
V V V V:
/• /• / /• /• /- / /- / /• /- /• /• / / /• /• / /- / / /• /•
\ v \ \ v v v \|
v
V
n
\, vv vv
V
n^
j
V
n
yv
V
n
W \ v v v v v v
n
WWW\i(?^^
•
n^UA«
1_J
AOZ
|_Jaos
UA0*
(p)
<*)
(6)
fMJ
o—
—
\.
i
;
.
j
.J
.
t
\
_
i
"
^
1
£
!
s
I
I
s-
j
<.
'
)
1
V
[
J
J
"1
i
(
i
j
j
i
i
;
i
fc
!*~
$-
i
-1.
_
i
£
1
•_
_
d
116
c
G
jf
i
C
D
(
I
I
^.
V
c
I
s
c
^
<>
;:-
_
g
P
c~*
-^_
"*%
(.
^
t
^.
\
^
2z
C^
_
c_
/
_
£
^„
^
^_
s~
^
t
^_
/•
v.~
•
s~
--.
V
~
_
/_
b-
V-
/'
^
s.
s-
d
q
>
o
C
CM
T$
d
d
^
r
a
a
8
o
2
2
117
s
c
c
C
:
c
C
<;
v
^
<"
s
<T
>
V
V
>
>
s,
^
^
.18
:j>
Fig. 38
The
sam
oscillograms of Fig.
37 taken with
u;
=
1.3
[7}=1
a'
Mra
=
0.4
Vo
=
1
7o
=
enlarged scale.
170°
=0
Fig. 38 i
(a)
=
0.1
i\
(d)
=
0.25
u1
(g) =5i'fl
(b)
=
0.1
l\
(e)
=
0.75
«£
(h)
=
(c)
=
0.1
r3
(0 =2ii
=
0.25
/}
F/£. J*//
Ud
=
1.3
U}=
1
a'
M^
=
0.4
?o
!
}'o
(a)
=
0.1
=
170"
=0
di\
i'.
(d)
=
0.5
,•;
(b)
=
0.1
ij
(e)=0.05^
(c)
=
0.5
/;
(f)
(g)=0.05
0.5
(h)
=
l
a'
=
=1
Vo
—L
rffc
urL
-77^
dira
=
0.05 —2
fVg. 55 Hi
Vrd
=
1.3
U',=
M^
=
0.4
*„
(a)
=
0.25
u;
(d)
=
0.25
«;
(g)
=
(b)
=
0.25
uj
(e)
=
0.25
u^
(h)
=
(c)
=
0.25
hJ
(f)
=
0.25
«&
1/>=1
a'
=
=1
Vo
Fzg.
.
170°
=0
0.25
-^
0.25 —^
38 iv
Urd
=
1.3
Mi
=
0.4
(a)
=
0.25
(b)
=
0.5
(c)
=
0.5
y0
Mre
(d)
=
M
(e)
=
rq
(f)=0.25^
0.5
0.5
170°
=0
^
(g)
=
^
(h)
=
0.25
-^
dtc
0.5 y
119
fl-. .9.
^
^
/
/
i
^i
^
v
>
s
>
>
/
/
s
/
/
^
/
V
/
c.
<'
>
/
/
r
/
>
120
-
<^
,r
j'
*"
/
'"1
c
y
^.
\
/
y
'/
<?i
:\
.so
c
f
!,
*
>
3
*
c
r
r
c
121
!
^
^
^
^
k
^
£
^
<
u
c
c
<J
c
f
d
4
a
122
c
(
c
(
(^
c
c
c
/
r
/
g
<1
I
c
!
;>
?
22
?
7
>
a
d
ti
123
Characteristic
9.3
curves
9.3.1 External characteristics
This shows the behaviour of the average value of the d.c. current
with respect to the
7dr,c-
current
lie.
applied voltage Urd
given by
is
=—
In
J
(f1 +
+
i£+
+
and with a!
as
I^.c.
parameter. The
i3+)dfr
o
This
Fig.
carried
was
for the
on
39 shows the
t^-Jd.c.
when
case
y
=
1 and
Uj
=
l.
for different values of a'
curves
parameter.
as
9.3.2 Current locus
The
procedure given
same
in section 6.4.3 to obtain the locus of the
fundamental component of the current in the
applied
in the
case
of the
three-phase
consideration the fact that the
phase voltage
fundamental component of the current
component of the voltage
when
y
=
1 and
Urf
machine works with
=
a
Fig.
as a
is not
as
a
case was
was
pure^sine
again
taken into
The
wave.
drawn with the fundamental
was
carried
on
for the
case
parameter/It is'noted that the
better power factor when it is
a', than with lower a! which is
works
was
reference. This
as
1 and with a!
single-phase
machine. Thus it
nearer
to the
operated with higher
region where the machine
generator.
40 shows the
9.4 Behaviour
family
of the
of curves where the machine works
air gap
flux
as a motor.
vector
The aim of this section is to show the behaviour of the resultant air gap
flux vector and the vector
case
of the
The flux
corresponding
controlled
three-phase
rpj linking the (/-winding was
to the armature reaction in
the
synchronous machine.
recorded with the time axis extended
in order to increase the accuracy of the measurement. In order to obtain
corresponding to that winding, the leakage flux was sub¬
xjjrd, then it was drawn on polar co-ordinates. Because this
the air gap flux
tracted from
oscillogram
measure
was
for the
The
same was
and
taking
respect
124
taken for the
angle
case
also made
on
speed, the time is
a
direct
\j/rq after subtracting the leakage flux
position effect of the ^-winding with
the flux
into consideration the
to the
of constant
y, which the rotor has moved.
cf-winding, namely
90° in advance.
udr
Fig. 39
nous
y
External characteristics of the
1
=
U}
=
controlled
synchro¬
1
1
a'= 110°
4
a' =160°
2
a'
=
120°
5
Jdr.c.
3
a'
=
140°
The resultant of these two
and shown in
Fig.
The factor 1.5 is
\]/Td and \prq
as a
a
41
curves
(curve
No.
curve
=
1
multiplied by the factor
1.5
was
plotted
3).
direct consequence of the mathematical definition of
function of
\j/[, \lir2, \jir3.
the direct axis flux vector with
Also
three-phase
machine
Curve No. 1 of
respect
Fig.
to the fixed axes
41 represents
co-ordinates.
No. 2 represents the quadrature axis flux.
In order to obtain the vector of the flux
originated by
the armature
125
Current
Fig. 40
locus
three-phase controlled synchronous
the
for
machine
7
=
U}
1
\
=
1
a'= 110°
4
a'=160°
2
a'
=
120°
5
lx
3
a'
=
140°
reaction,
we
minus the leakage flux. Thus
point
A
1
linking the winding rfwhen the armature
equals the total flux of the field winding
had to draw the flux
reaction is not present, which
The
=
on curve
No. 3
we
obtained
corresponds
Thus the armature reaction at this instant is
126
curve
to the
No. 4 of
point
B
Fig.
41.
on curve
No. 4.
represented by the vector BA.
41
Fig.
The flux vector
controlled
synchronous
linking the different coils of
drawn with respect to the fixed
C/dr
=
U}
0.16
a'
7=1
the
three-phase
machine and the armature reaction flux vector
linking
linking
axes
=
1.5
=
115°
the d-coil
1
=
the flux vector
2
=
the flux vector
3
=
the total air gap flux vector
4
=
the total air gap flux vector without the armature reaction
5
=
the armature reaction flux vector
This
was repeated on
corresponding points
the
different
on
curve
<?-coil
points
on curve
No. 3 and
No. 4,
deducing
determining the
the vector of the
armature reaction. Hence vectors were then drawn from the
the
curve
No. 5
vector with
was
respect
to
obtained
the fixed
representing
origin 0, and
the armature reaction flux
axes.
127
d
0.1 per unit of the flux
0
Fig.
moving
V\
v
The armature reaction flux vector drawn with respect to the
42
=
=
axes
d,
q
0.16
Urf
1
a'
=
1.5
=
115°
It is also of interest to draw that vector with respect to the movable
d and q. This
every vector
Thus the
was
was
curve
again deduced
moved backwards
shown in Fig. 42
from
by
curve
the
angle
No. 5 of
axes
Fig. 41, where
that the rotor has moved.
obtained.
was
9.5 Transient behaviour
The effect of sudden
torque Mra
Fig.
on
the different
43i...43 iv
1.3 to 1.1. The
are
much
change of the applied voltage Urd and the machanical
gives
oscillograms
the effect of
oscillograms of the
greatly
waves
affected than those
time constant tries to
keep
of the machine
were
on
128
seen
on.
in the quadrature axis direction
the direct axis because the field
the flux in the direct axis constant.
The effect of the sudden increase of the mechanical torque
to 0.7 is to be
carried
suddenly decreasing the voltage C/J from
in Fig. 44i...44iv.
Mra
from 0.4
Fig. 43
The effect of sudden decrease of the
oscillograms
Wf
=
of the
three-phase
a'
1
=
controlled
170°
applied voltage Ud
synchronous machine
Mra
Ud is suddenly reduced from 1.3
=
0.4
to 1.1
Fig. 43 i
(a)
=
0.1
i\
(d)
=
0.25
«;
(g)
=
5
(b)
=
0.1
r2
(e)
=
0.25
i}
(h)
=
0.125
0.1
C3
(f)
=
2fD
i;
(d)
(g)
=
(c)
=
fQ
Urd
Fig. 43 ii
(a)
=
0.1
=
0.5
i\
0.05
di\
—-
dtc
.
(b)
=
0.1
ij
(e)
=0.05-^
(c)
=
0.5
i'd
(f)
=
ut
(d)
=
-
(h)
=
«;
(g)
=
»r„t
(h)
=
0.125
Ua
dia
0.05 —q-
dtc
Fig. 43 Hi
dVa
(a)
=
(b)
=
0.25
urfi
(e)
(c)
=
0.25
«;
(f)=0.25^
M;
(d)
=
0.5
^
(g)
^
(h)
0.25
0.25
0.25
0.25 -^
dtc
0.125
£/£
dtc
F/£. 45/v
(a)
=
0.25
(b)
=
0.5
ft
(e)
=
0.5
(c)
=
0.5
Vq
(f)
=
0.25
—-
dtc
==
=
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10. Summary
and conclusion
This thesis deals with the
investigation of
thyratron controlled
a
syn¬
chronous motor.
The motor
picture
directly
behaviour,
of its
experimentally, but
examined
was
measurements made
on
e.g. it
it
was
not
possible
found that the
was
the machine do not
give
a
complete
to measure the instan¬
electromagnetic torque, or the currents in the damper
possible to investigate the behaviour of each quantity
the machine if it is mathematically analyzed. Since the syn¬
taneous value of the
windings.
belonging
It is
to
chronous machine is in
the
thyratrons
complexity,
a means
it
The system
found
was
solving
of
was
general complicated, and since the insertion of
silicon-controlled rectifiers in its
or
this
more
stationary
transformation
has the
solved for the
It
was
and then
advantage
then
especially
steady
taneous value of the
the
thyratrons
the
on
to take
the currents
sient and the
transformed
were
(d, q) moving
into
Oscillograms
electromagnetic
curves
was
studied for
a
a
axes.
were
the
(a, /?)
one.
This
also taken for the instan¬
torque and for the
were
steady states.
calculated, also
voltage
across
the loci of the
plotted against
were
that
parameter. The behaviour oftheairgap
certain
operating point
was
the
and another with respect to the
axes
The stator
Thus the whole
nonlinear
a
reaction flux vector
stationary
on
quantities of interest,
damper windings in the tran¬
fundamental component of the stator currents
of the stator voltage with a'as
cases.
first
at
for all the
in both the
in both the transient and
The external characteristic
flux vector
as
the necessary number of multipliers.
oscillograms
flowing
states.
linear and
a
of greatly reducing
possible
analog computer
use an
single- and the three-phase
separated
was
stator increases this
problem.
currents, voltages and fluxes
axes
convenient to
deduced. This
was
drawn
and the armature
once
with respect to
moving axes. The
applied voltage on the different
oscillograms show how the motor functions
effect of the sudden variation of the
oscillograms
as a
was
generator for
shown. The
short time and then back to the motor action
of
through
applied d.c. voltage Ud. This is most obvious
the electromagnetic torque and that of the
considered.
The effect of the sudden variation of the
a
the sudden decrease of the
when the
current
iq
oscillograms
are
mechanical torque
was
also studied.
139
The
pulses controlling
voltages taken
from
machine. It is
source,
so
the
proposed
that the
synchronous machine in
from
that these
speed
our case were
tachometer mounted
a
pulses
may be varied
on
may be obtained from
by varying
the
obtained
the shaft of the
separate
a
of that
frequency
source.
publication made by Prof. Ed. Gerecke [21], the problem was
making some simplifications. All the resistances in the
a.c. side were neglected and the inductance in the d.c. side was assumed
to be infinitely large. However, an agreement to a great extent can be
In the
calculated after
oscillograms obtained in this thesis and those got in
paper. The angle a' in this thesis equals (60 + \j/) in that
noted between the
the mentioned
paper. The simulation of the motor in this thesis
types of computers, the first
PACE
one.
ing
of
15
bistable
a
The simulation
circuit
simulating
on a
was
executed
on
"Donner" type and the second
two
on a
on
the Donner computer demanded the build¬
the
thyratron effect. This contained 18 gates,
multivibrators,
15
Schmitt
triggers
and two
saw
tooth
were
made
generators.
The comparators
use
140
already built in
of to simulate the
the PACE
thyratrons.
analog computer
ZUSAMMENFASSUNG
Untersuchung eines kommutatorlosen Gleichstrommotors, bestehend
einer
aus
Drehstrom-Synchronmaschine und
Analogrechner.
einem
Wechselrichter,
durch Simulation auf einem
Zweck der Arbeit ist die
Untersuchung aller Strome und Spannungen
sowie des Drehmomentes eines kollektorlosen Gleichstrommotors. Dieser
besitzt eine normale
dreiphasige,
in Stern
geschaltete Statorwicklung
Erregerwicklung und zwei
Langs- bzw. Querachse angeordneten Dampferwicklungen. Die
und ein Polrad mit ausgepragten Polen, einer
in der
Speisung erfolgt von einer Gleichspannungsquelle aus iiber eine Glattungsinduktivitat und iiber eine dreiphasige, mit sechs steuerbaren Ventilen (Thyratronen oder Thyristoren) ausgeriistete Bruckenschaltung.
Zunachst wurde ein 4-kW-Motor
der Ventile
erfolgte
von
experimentell untersucht. Die Steuerung
angebrachten drei-
einem auf der Motorwelle
phasigen Tachogenerator aus. Es wurden die Oszillogramme aller zuganglichen Strome und Spannungen im Ein- und Dreiphasenbetrieb aufgenommen und ferner das Drehmoment elektronisch gemessen. Ferner
wurden alle fur die anschlieBenden
Berechnungen notigen
Daten der
Maschine ermittelt.
Fur die theoretische
Untersuchung wurde zunachst das Dreiphasen-
system des Stators in ein ruhendes Zweiphasensystem umgewandelt und
dieses nach der Zweiachsentheorie in ein mit dem Polrad synchron rotierendes
Zweiachsensystem iibergefuhrt.
Uber die Kurvenform der Strome
Spannungen wurden keine Annahmen gemacht. Ferner wurden der
Erregerstrom sowie die beiden Dampferstrome in die Rechnung einbeund
zogen. Zudem
die Verschiedenheit des
magnetischen Widerstandes
Querachse
Streuungen und
eine veranderliche Drehzahl zu beriicksichtigen. Es handelt sich also um
ein hochgradig nichtlineares System, weshalb die Laplace-Transformation
in der
Langs-
waren
und
sowie die verschiedenen
nicht verwendet werden konnte. Fur die Koordinatentransformationen
wurden die benotigten Matrizen ermittelt.
bedingungen der Ventile formuliert werden.
Ferner muBten die Schalt-
Die
Nachbildung wurde alsdann auf total sechs kleinen Analogrechnern
(Donner, Kalifornien) vorgenommen. Da die Zahl der Multiplikatoren
auf acht beschrankt war, muBte die
Programmierung
unter
Verwendung
141
der vorhandenen Mittel
dingte.
Fur die
durchgefuhrt werden,
Erzeugung
der
Zundimpulse
was
Differentiationen be-
wurde ein transistorisierter
sechsphasiger Impulsgenerator gebaut. Es gelang dann, sowohl im Einphasen- wie im Dreiphasenbetrieb die Oszillogramme aller Spannungen
und aller Strome, ja sogar der beiden Dampferstrome, aufzunehmen. Die
Obereinstimmung mit den experimentell ermittelten war befriedigend,
jedoch zeigten sich zufolge der Differentiation einige Unschonheiten.
Daher wurde das Problem nochmals auf dem
von
der ETH inzwischen
angeschafften groBen Analogrechner «PACE231-R»programmiert, woes durch Umformung des Gleichungssystemes gelang, ohne irgendwelche Differentiationen auszukommen. Es ergaben sich nun sehr saubere
Oszillogramme aller Strome und Spannungen sowie des elektromagne-
bei
zeigt neben dem konstanten Wert einen
Statorfrequenz leicht pulsierenden Anteil. Ferner wur-
tischen Drehmomentes. Dieses
mit der dreifachen
Oszillogramme ebenfalls bei transienten Zustanden, wie plotzplotzlicher Anderung des Lastmomentes und bei
schrittweiser Veranderung der speisenden Gleichspannung, ermittelt. Es
zeigte sich dabei, daB die Maschine kurzzeitig als Generator arbeitet und
den alle
lichem Belasten, bei
Gleichstrom
142
an
die
Speisequelle
zuriickliefert.
Literature
[1] Park,
R. H.: Two-reaction
alized method of
theory
of
synchronous machines, gener¬
Transactions, vol. 48, 1929,
Part I. AIEE
analysis.
pp. 716-727.
[2] Concordia, C: Synchronous machines, theory and performance.
John Wiley and Sons, Inc., New York 1951.
[3] Laible, T.: Die Theorie der Synchronmaschine im nichtstationaren
Betrieb.
[4]
Springer-Verlag,
1952.
White and Woodson: Electromechanical energy conversion. John
Wiley
Sons, Inc.; New York 1959.
and
[5] Kovacs,
K.
P., und RAcz, I.: Transiente Vorgange in Wechsel-
strommaschinen. Band I.
Verlag
der
Akademie der
ungarischen
Wissenschaft, Budapest 1959.
[6] Hannakam,
Dynamisches
L.:
Verhalten
von
synchronen
Schenkel-
polmaschinen bei Drehmomentstofien. Archiv.fiir Elektrotechnik,
[7]
Band XLIII, Heft 6,
1958, S. 402-426.
AIEE Test codes for
synchronous machines,
No.
503, June 1945.
[8] Waring, M. L., and Crary, S. B.: Operational impedances of
a
machine. General Electric Review, 35, 1932, p. 578.
synchronous
[9] Nurnberg, W.: Die Priifung elektrischer Maschinen. Berlin 1940.
[10] Badr, H.: Primary-side thyratron controlled three-phase induction
machine. Diss. No.
3060, ETH, Zurich.
P.: Simulation of
[11] Coroller,
a
Brown Boveri
logue computer.
synchronous machine
with
an ana¬
Review, vol. 46, No. 5,1959, pp. 299-
306.
[12] Willems,
dynamique des reseaux interconnected.
Sciences appliquees, Universite Libre de
P.: Sur la stabilite
Une These de Doctorat
en
Bruxelles, 1956.
[13] Fifer,
S.:
Analogue computation. McGraw-Hill, Inc.,
New York
1961.
[14]
Johnson:
Analog computer techniques. McGraw-Hill, Inc.,
New
York 1956.
[15]
Korn and Korn: Electronic
analog computers. McGraw-Hill, Inc.,
New York 1952.
[16]
Millman andTAUB: Pulse and
digital
circuits.
McGraw-Hill, Inc.,
New York 1956.
143
[17] Pressman, A.: Design of transistorized circuits for digital computers.
John F. Rider, Inc., New York 1959.
[18]
SEV
0192, 1959: Regeln und Leitsatze fur Buchstabensymbole und
Zeichen, 4. Auflage, 1959.
[19] International
Electrotechnical
dations for mercury
[20]
Baumgartner:
arc
Commission
Dynamisches
Verhalten
reaktiven Verstarkermaschinen mit
gliedern.
ETH
[21 ] Gerecke,
Ed.
:
von
Recommen¬
84, 1957.
Regelkreisen
Begrenzungs-
und
und
Hysterese-
Zurich, Promotionsarbeit Nr. 2916, 1959.
Synchronmaschine
Technik 3, 11, 758-766 (1961).
144
(IEC),
converters, Publication
mit
Stromrichterbelastung.
Neue
Curriculum Vitae
born in Alexandria, Egypt, on the 25th of September 1933.1 attended
primary school during the years 1942-1946 and the secondary school
during the years 1946-1951, both in Alexandria. I passed the maturity
I
was
the
examination in
July
1951. In October 1951, I
began
my studies at the
Alexandria University, and in June 1956, I
was
Engineering,
in
awarded the B.Sc. degree
Electrical Engineering. From October 1956
until February 1958,1 have worked at the Ministry of Industry in Cairo.
In April 1958, I began my studies at the Institute of Automatic Control
and Industrial Electronics of the Swiss Federal Institute of Technology
at Zurich under the supervision of Prof. Ed. Gerecke. In November 1959,
I passed the admission examination to the doctorate work. This thesis
was carried out during the period from 1960 until June 1963. I passed
Faculty
of
the doctorate degree examination
on
the 19th of July 1963.
145