Appendix
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[[PDF_PAGE:28]]
22
Report of Charles P. Steinmetz
APPENDIX
Synchronous Operation
A
Consider the case of two
alternators or groups of alternators
such as station sections, which are running in synchronism with each
other, that is, have the same frequency f, but are connected together
while out of phase with each other by angle 2w.
That
is, the one
alternator
has
the
voltage
phase
(<f>
to),
the
other
the
voltage
phase (0+w).
We may assume the
alternators
as
of equal
voltage,
since
a
voltage difference superposes on the synchronizing energy current
due to the phase difference, a reactive magnetizing current due to
the
voltage
difference
without
materially
changing
the
energy
relations.
The EMFs of the two alternators then may be represented by:
ei = E cos (0
co)
1
e2 = Ecos (0+co)
/
(1)
and the resultant voltage in the circuit between the alternators then is :
e = ei
e 2
= E cos
\
(<f>
co)
cos
(</>+ co)
[
= 2E sin
co sin
(2)
and the interchange currentwbeteen the alternators is:
2E
.
i =
sin
co sin
(<j>
a)
(3)
where:
z = r2+x
2
is the impedance of the circuit between the two alternators, and the
phase angle a is given by:
x
tan a = -
r
and:
r= resistance
x = reactance
of
the
circuit
between
the
alternators
(including
their
internal
resistances and reactances).
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[[PDF_PAGE:29]]
Report of Charles P. Steinmetz
23
The power of one of the two alternators then is given by:
2E
2
=
sin
co sin
(d>
a) cos
(<f>
co)
z
E
2
f
1
=
sin
co sin
{ (20
a
co)+sin
(co
cm
Z
I
J
E
2
E
2
E
2
=
sin
co sin
(2<f>
a
0))+^- cos a
jr- cos (2co
a)
(4)
^7
^7
The phase angle
co of the EMF is not constant, but pulsates with
approximately constant low frequency, the frequency of the beat,
and decreasing amplitude.
co =co oe
= maximum value
of the phase
angle,
then may
approximately
represent
the
gradually
decreasing
amplitude
of
the phase angle, where a = attenuation of the beat or oscillation, and
-at
.
...
co=a>oo
sin pc/>
(5;
would approximately represent the instantaneous value of the phase
angle
co where:
pf= frequency of the beat, or the periodic variation of the phase
angle.
[In the derivation of equations
(3) and
(4),
co has been assumed
as constant.
As
co is not constant, but by (5) a function of <, addi-
tional terms appear in equations
(3) and
(4).
Since, however, the
frequency of variation of
co is very low compared with the frequency
of
<j>
: p = a small quantity ; these additional terms in (3) and (4) are
small, and equations (3) and (4) are correct with sufficient approxima-
tion, especially in the present case, where we are essentially interested
in the magnitude of the power relations.]
In equation (4), the first term:
E
2
.
Pi
sin
co sin
(2c>
a
co)
is of double the frequency
2f.
It thus does not represent energy
transfer between
the two
alternators,
but merely
represents
the
energy storage and return, twice per cycle, occurring in any inductive
circuit.
It thus is of no further interest.
The second term
p"i=- cos a
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[[PDF_PAGE:30]]
24
Report of Charles P. Steinmetz
gives, substituting cos a = -
;
z
E
2g
P"i=2?
2
where g is the conductance of the circuit.
That is, this term is the
energy loss in the conductance, that is, the resistance of the circuit,
and thus also is of no further interest.
The third term:
p'Y=-!|cos(2a>-a)
is of the low frequency of the beat, or the current fluctuation between
the two
alternators:
pf.
It thus represents the energy transfer
between the two alternators, during their periodic oscillations,
or, resolving the last equation:
E
2
E
2
p'Y=
TT sin a sin 2a>
=- cos a cos w.
2z
2z
The second term:
E 2
p'Y'=
- cos
a. cos
a>
has the same sign for negative w, that is, when the machine is lagging,
as for positive w when the machine is leading, thus it represents no
energy transfer between the machines.
The synchronizing power, or energy transfer during the synchro-
nizing oscillations of two alternators, which are out of phase but in
synchronism, thus is given by the expression:
E
2
P=- sin a sin 2co
(6)
Thus, the synchronizing power p, is a maximum, and
is :
_E
2
.
for a = 90 degrees, that
is,
if the resistance r of the circuit between
the alternators is negligible compared with the reactance.
The synchronizing power p =
for a = 0, that is, in the (theoretical)
case, when the circuit between the alternators contains no reactance,
but only resistance.
For phase angles w up to 45 degrees, that is, phase displacements
between the two alternators up to 2 w = 90 degrees, the synchronizing
power increases ; beyond this it decreases again and becomes zero for
2w=180 degrees phase displacement.
[[END_PDF_PAGE:30]]

[[PDF_PAGE:31]]
Report of Charles P. Steinmetz
25
The average value of p may be approximated by:
E 2
.
2 (l-cos2a>
)
E 2
.
avg. p=^r
sin a
- -- =- sin a (1
cos 2a>
)
2Z
7T
COo
7TCOZ
where
coo denotes the maximum value of
co
and as the duration of one half cycle of oscillation
during which the
power transfer remains in the same direction
is given by equation
(5) as:
p$ = 27rpft = 7r
that is,
t==
2p~f
The energy transfer between the two machines, during each half
cycle of oscillation, is given by:
W =
avg. p.
(8)
F 2
sin a (1
cos 2&>
)
2 TrpfcooZ
in
this, E
is the maximum value of the voltage of each machine.
Denoting by :
F
- E
E ~72
the effective value of the machine voltage, gives the effective values :
Resultant voltage:
e = 2E
sinco
(2
1
)
Current:
2F
io=
- sin w
(3
1
)
z
Power transfer:
F
2
p=
sin a sin 2co
(6
l
)
z
Energy transfer during each half cycle of oscillation or beat:
W= ^-sin a (1-cos 2 Wo )
(8
1
)
TrpfwoZ
where :
co=cooo
sin p<
(5)
is
the
angle
of phase
displacement
of
either machine, from
the
average ;
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[[PDF_PAGE:32]]
26
Report of Charles P. Steinmetz
z= Vr 2+x
2
t an a = -
r
r= resistance of circuit.
x = reactance of circuit.
p= frequency ratio of beat or oscillation, that is,
pf= frequency of oscillation, and
E = effective value of machine EMF.
As seen from the equations, during each complete cycle of the
oscillations, of frequency pf, the current i twice rises and falls, thus
reaching two maxima, and the power p twice reaches a maximum
and twice reverses, so that the energy W flows one way during half
the cycle, and in the opposite direction during the other half cycle.
The frequency of the rise and fall of the current thus is 2 pf.
Figure 1.
I and II show the curve i and voltage ei of the oscilla-
tion,
for
the
(exaggerated)
value p = .l,
for
o> =45
degrees and
wo = 90 degrees.
B
Consider now the case that two alternators, or groups of alter-
nators such as station sections, are connected together while different
from each other in frequency by 2s, that
is, one alternator has the
frequency (1
s)
f, the other the frequency (1+s)
f.
We may again assume the alternators as of equal voltage, since a
voltage difference merely superposes on the synchronizing energy
current a reactive'magnetizing current, without materially changing
the energy relations.
The EMFs of the two alternators then may be represented by:
CI=E cos (i
s)j<
\
e 2 = E cos (1+s)
<t>
J
(9)
The resultant voltage in'the circuit between the two alternators then is
:
e = ei
e 2
= E<cos (1
s)
<p
cos (1+s)
<f>\
= 2E sin s0 sin
<
(10)
[Assumingjnow that s is a small quantity (just as we assumed in
A, that p is a^small quantity), that is, that the two alternators have
nearly the samejfrequency.
The change of sin
s<
then is slow com-
pared with that of sin<, and for all phenomena of the frequency
f,
sin
s<j> may be assumed as constant, and the reactance of the circuit
[[END_PDF_PAGE:32]]

[[PDF_PAGE:33]]
Report of Charles P. Steinmetz
27
may be assumed as the same, x = 27rfL, for both component EMFs,
ei and e 2 that
is, for both frequencies (1
s) f and (1+s)
f.]
The interchange current between the alternators then is:
i =
sin
s<t> sin (0
a)
z
.
where:
z= Vr 2+x
2
tana=-
(11)
With regards to the EMF of one of the alternators, for instance,
ei this current always lags.
Its lag is 90 degrees when the current is
a maximum.
With decrease of current, the lag decreases from 90
degrees in the one, and increases in the next beat, and approaches in
phase respectively in opposition, when the current
is a minimum.
The power factor thus varies from zero at maximum
current,
to
unity at zero current, and its average thus is low.
Fig.
1 shows as
Curve III the relation of ei to i for the exaggerated values s = .09.
The power of one of the two alternators then is given by :
2F" 2
=
- sin
s<
sin ($
a) cos (1
s)
<f>
E
2
f
|
=
sin s0
<
sin [(2
s)
<f>
a]+sin
(s<
a)
>
z
I
J
TT 2
TT 2
TT 2
=
sin
sin [(2
s)
<4
a]-}-s- cos a
^- cos (2sd>
a)
z
2z
2z
The
first term again
is the double frequency term representing
the energy storage by inductance; the second term
is the power
consumed by the resistance of the circuit.
Neither thus represents
energy transfer between the alternators.
The third term
:
E2
(sj,_
\
(i2\
is
a
slow
pulsation
of
energy,
which
alternately
accelerates
the
machine and thus tends to bring it nearer to synchronism, and then
retards it again.
Usually, it is approximately: a = 90 degrees, that is, the reactance
is large compared with the resistance, and equation
(12) then be-
comes :
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[[PDF_PAGE:34]]
28
Report of Charles P. Steinmetz
E
2
p =
2^sin2s<
During each cycle of the frequency sf, of the
slip from synchro-
nism or average frequency, the amplitude of the current
i thus twice
becomes zero and in phase, and twice reaches a maximum, when the
alternators are in opposition, and the power p four times reaches a
maximum and four times becomes zero and reverses, twice when
the current comes into phase with the EMF, but the current becomes
zero, and twice when the current is a maximum, but in quadrature
with
the EMF, and the power thus becomes
zero.
The power
transfer between the alternators thus reverses four times per com-
plete cycle of slip, sf, that is, is of the frequency 2sf, with two positive
and two negative maxima.
The average value of the power is:
2 = Ef
7T
7TZ
and as the duration of one quarter cycle of slip is t =
j-p, the energy
transfer between the two machines, during a quarter cycle of slip
thus is:
1
2
-S55
There is thus that difference between the slipping of alternators
past each other out of synchronism, and the oscillation of the alter-
nators against each other at synchronism
(A), that in the slipping
the power fluctuation and the reversal of the energy is of twice the
frequency of the current fluctuation, while in the oscillation of the
alternators against each other at synchronism, the power fluctuation
or reversal of energy flow
is of the same frequency as the current
fluctuation.
If two alternators
are connected together while out of synch-
ronism, and slowly
slip past each other, during each half cycle of
slip, or beat, while the two machines EMF pass from in phase to in
opposition to in phase again, a periodic energy transfer takes place.
During one quarter cycle of slip (that
is, while one alternator EMF
slips behind, the other pulls ahead of the minimum frequency by
one quarter cycle, and the two alternator EMFs thus
slip against
[[END_PDF_PAGE:34]]

[[PDF_PAGE:35]]
Report of Charles P. Steinmetz
29
each other by one quarter cycle), the alternators are partly in phase
with each other, and the slower machine receives energy from the
faster machine.
The two machines are thereby brought nearer to
each other in speed, pulled towards synchronism.
During the next
quarter
cycle of
slip, however,
the two
alternators
are partly
in
opposition, and the faster machine receives energy from the slower
one.
The faster machine then speeds up, the slower machine slows
down, and the two machines pull apart again, by the same amount
by which they pulled together in the preceding quarter cycle of slip
(if their EMF
is constant).
Thus the two machines can pull into
step only if the energy transferred during one quarter cycle of slip,
W, is larger than the energy required to speed up the momentum M
of the machine, to full synchronism.
Due to the energy transfer W between the machines, resulting
in an alternate speeding up and slowing down, the slip s is not con-
stant, but pulsates periodically, between the minimum value s
Si,
at the end
of the quarter
cycle during which the machines pull
together, and
beginning
of
the
quarter
cycle
during which
the
machines pull apart, and a maximum value s+Si, at the end of the
quarter cycle, during which the machines pull apart, and beginning
at the quarter cycle during which the machines pull together
where
Si is the amplitude of the pulsation of slip.
As the energy required
to accelerate the momentum M of the machine by the speed 2s i
is
4s jM, it follows:
W
E
2
l
16
7rsfzM
is the amplitude
of the speed fluctuation
of the two
alternators
during their slipping past each other, out of synchronism with the
slip
s.
Si = s gives the minimum slip s
Si = 0, that is, the machines pull
into synchronism.
The maximum
slip
Si from which the two machines
pull into
synchronism with each
other, thus
is given by
substituting
si = s
in (14), as:
E
So=
[[END_PDF_PAGE:35]]

[[PDF_PAGE:36]]
30
Report of Charles P. Steinmetz
So thus is the limit of synchronizing power.
Substituting again the effective value of EMF, E
, for the maxi-
mum value E, by:
JR
Eo= V2
gives the effective values:
Resultant voltage:
e = 2E
sins4>
(10
l
)
Current:
2F"
io = ^-sins<
(Hi)
z
Power transfer:
p =
cos (2s0-a)
(12
1
)
X
Energy transfer during one quarter cycle of sUp:
Amplitude of pulsation of slip:
Critical Slip, from which
the machines pull each other into synchro-
nism, or limits of synchronizing power:
These expressions are very similar to those of A, with
s<
taking
the place of
, and 2s the place of p.
C
With two machines out of synchronism with each other by a
greater speed difference 2s, than that, from which the machines can
pull each other into synchronism within one quarter cycle of slip,
from the equations of B it would follow, that the machines can never
pull each other into synchronism,
if the voltage E
is constant, but
must indefinitely continue to
slip past each other, coming nearer
together during one quarter cycle of slip, and dropping apart again
by the same amount during the next quarter cycle of slip.
This, however, is under the assumption, that the machine EMF
E
is constant.
In reality, however, E
is not constant, but varies
periodically with the same frequency as the current fluctuates.
The
[[END_PDF_PAGE:36]]

[[PDF_PAGE:37]]
Report of Charles P. Steinmetz
31
current in the circuit between the machines and thus the armature
reaction in the machine varies in amplitude and in phase difference
against the machine voltage, and the machine voltage varies with
the amplitude and the phase of the armature reaction.
Consider,
as approximation,
the armature reaction as propor-
tional to the quadrature component of the current.
The EMF of
the machine would then be expressed by an approximate equation of
the form:
E = E
|
1
c sin
s<
sin
<f>
\
where
<f> is the phase angle between the current and the EMF and
sins0 represents the amplitude of the current pulsation, by (II
1
)
it is, however, by (9) and (11):
<p=(<f>
a)
(1
s)
90 degrees
= s0
a+90 degrees
thus:
f
1
E = E
| l+c sin s0 cos (s0
a)
j>
(16)
Substituting
(16) into the expression of the power of the alter-
nator (12
x
), the equations
still remain alternating, that
is, there
is
no resultant synchronizing power, but equal positive and negative
values of power alternate.
However, (16) assumes that the magnetizing effect of the armature
reaction
is instantaneous, that
is, that the EMF E at any moment
is the value corresponding to the armature reaction existing at this
moment.
This, however, is not the case, and the armature reaction
is
not
instantaneous,
but
requires
an
appreciable
time,
several
seconds, to develop, and the magnetizing or demagnetizing effect of
the armature reaction on the voltage therefore materially lags behind
the armature reaction.
Let then a = angle of lag of the voltage change behind the armature
reaction which causes it.
It is then
f
1
E = E
{ l+c sin s0 cos (s0-a
<r)
\
(17)
I
J
and, substituting (17) into (12
1
), give the power transfer between the
machines :
E
2
f
1
2
p=
- cos (2s0 = a)
\ 1+c sin s0 cos (s0
a
a)
\
z
[[END_PDF_PAGE:37]]

[[PDF_PAGE:38]]
32
Report of Charles P. Steinmetz
or approximately, since c is a small quantity:
p =
cos (2s$
a) -\
cos (2s0
a) sin s$ cos (s0
a
a)
z
z
The first term:
E
2
p' = -
cos (2s$
a)
is
the
slowly
alternating
energy
transfer between
the
machines
which causes
their speed to
fluctuate, but does not permanently
bring them nearer to each other, that
is,
exerts no synchronizing
power.
The second term :
p" =
cos (2s0
a) sin s<
cos
(s<
a
a)
z
f
=
- cos (2s0
a) | sin (a
- sin (a+o-) cos (2s0
a)+^^-|sm(4s0
2a
<r)
sina-
The first two terms of this expression :
TT
2
sin (a+a) cos (2s<
a)
sin
4s</>
2a
z
and
cE
2
2z
are slowly
alternating, thus represent no permanent
acceleration,
that is, no synchronizing power.
The third term however:
cE
2
.
P ~2z~ Sm *
'
'
is constant, that
is, represents a continuous synchronizing power,
which steadily pulls the machines together, until their speed differ-
ence 2s has become small enough to pull the machines suddenly into
step.
If thus two alternators or station sections, are considerably out
of synchronism with each other, they continue slipping past each
other, with large fluctuating currents flowing between them, and the
speeds of the machines fluctuating with the fluctuation of the current.
These currents do not decrease in amplitude, but remain of practically
[[END_PDF_PAGE:38]]

[[PDF_PAGE:39]]
Report of Charles P. Steinmetz
33
constant value, but their period of fluctuation gradually gets slower,
that
is,
the
fluctuation gradually becomes
slower,
while currents
slowly pull the machines nearer into synchronism with each other,
that is decrease their frequency difference, until the critical frequency
2s
is reached (where the acceleration during a quarter cycle of slip,
2si, reaches full synchronism.)
Then the machines suddenly drop
into synchronism, but oscillate in phase against each other, with an
approximately constant frequency of oscillation, but with a current
fluctuation,
which
steadily
(and
usually
rapidly)
decreases, until
steady conditions of speed, current and voltage are reached.
The armature reaction of the alternator
is represented by the
difference of the synchronism reactance x and the true reactance xi,
that is, by an effective reactance of armature reaction.
X2 = Xo
Xi.
The co-efficient c in the synchronizing power, p
(18),
is that
fraction of the reactance of the armature reaction x 2 , which appears
during the short time of the current fluctuation.
Thus c is the larger,
the slower the fluctuation, that
is, the
less
s.
In other words,
c
increases
with
decreasing
slip,
that
is,
increasing
approach
to
synchronism.
Inversely, a
is a maximum and practically 90 degrees for large
values of s, where the voltage fluctuation lags practically 90 degrees
behind the fluctuation of the armature reaction, and decreases with
decreasing
s, that
is, increasing approach to synchronism,
c sin a
and thus the synchronizing power p
(18), thus should be a maximum
at some moderate slip
s, and decrease for larger as well as smaller
sh'ps.
Assuming that
it takes
to seconds for the
field to build up to
correspond to the armature reaction.
With the current fluctuating
with the frequency 2sf, and assuming that the magnetizing effect of
the
armature
reaction
is
sinoidal
which at best
is but
a very
crude approximation, it would be:
1
~4sft
and:
thus:
[[END_PDF_PAGE:39]]

[[PDF_PAGE:40]]
34
Report of Charles P. Steinmetz
However, secondary
effects occur and more or less modify the
value po such as the
effect
of secondary currents, induced in the
field structure by that component of the armature current which is
due to the EMF of the other machine, and which gives an induction
motor torque, tending to pull the machines together into
synch-
ronism.
D
Consider two alternators or groups of alternators such as station
sections of the same terminal voltage, connected with each other
through an impedance
z, and in synchronism with each other.
If
then the load distribution between the alternators differs from the
distribution of their driving power, electric power is transferred over
the impedance z, current flows and a phase displacement 2co occurs
between the two sides of the reactor z.
In this case, the phase angle w is constant, and not periodically
fluctuating as in A, but varies with changes of distribution of load ;
the equations, however, are the same as in A, except that now w is
constant, and the voltages ei and e 2 are the terminal voltages.
It thus is,
Current in the impedance:
2E
.
1 =
sm
a) sm
(<
a)
JE
Voltage across the impedance:
e = 2E sin
co sin
<
Power transferred over impedance:
E
2
P =
2~
tcos a
cos (2co
a)]
where E is maximum value of the terminal voltages:
CI=E cos
(</>
w)
e 2 = E cos (0+w)
and a the phase angle of the impedance.
If the impedance z
is a reactance, that
is, the resistance
r,
is
negligible:
r = 0, it is a = 90 degrees, and:
2E
.
i =
sm
o> cos
<(>
x
e = 2E sin
co sin
<
E 2
=
sm
[[END_PDF_PAGE:40]]

[[PDF_PAGE:41]]
Report of Charles P. Steinmetz
35
or, substituting the effective values:
E = E
V2:
2E
.
ID=- sin w
e = 2E
sin w
p=
sin 2w
x
The power p thus is zero for
co = 0, increases, reaches a maximum:
E
2
Dm
X
for w = 45 degrees, or 90 degrees phase displacement between the
Iternators, and then decreases again to zero at w = 90 degrees, or
opposition.
At maximum power transfer, it is:
E
V2
_
1m
= E
V2
E
The foremost
difficulty, and uncertainty
in the application of
the preceding equations, is found in the selection of the proper values
of the machine EMF E
.
E
is not the terminal voltage; by slipping
past each other without external impedance, the terminal voltage of
the
alternators
goes down
to
zero.
Neither
is E
the "nominal
induced voltage," as this has no actual existence, but is the voltage
which would be induced by
the
field
excitation
if the saturation
curve of the machine continued as a straight line.
It appears to me
that E
must be considered as the actual induced voltage, that
is,
the voltage induced by the actual field
flux, that
is, the field flux
due to the resultant of field excitation and armature reaction. The
armature
reaction,
however,
fluctuates with the current between
zero and a maximum, while the actual field flux
is practically con-
stant,
since the magnetic
field cannot follow the
relatively rapid
fluctuation of armature reaction.
The
magnetic
effect
of
the
armature
reaction
is represented
electrically
in
the
synchronous
reactance
XQ.
The
synchronous
reactance thus consists of a true self-inductive reactance
Xi, which
is instantaneous, and an effective reactance of armature reaction x,
[[END_PDF_PAGE:41]]

[[PDF_PAGE:42]]
36
Report of Charles P. Steinmetz
which requires appreciable time to develop, and does not correspond
to any real magnetic flux.
In turbo-alternators, x 2 usually is very much larger than xi.
Electrically, the actual induced EMF thus should be the nominal
induced voltage e
, which corresponds to the
field
excitation,
less
the reactance drop of the average current in the effective reactance
of armature reaction, x 2 .
If then
I = maximum effective value of the fluctuating current,
the average current is
~, and the actual induced voltage thus is:
m
=
~-
It
is,
however,
in two
alternators connected together out
of
synchronism, through an additional reactance:
2E =I (2Xl +x)
where x is the additional reactance through which the alternators of
actual induced voltage E
and true self-inductive reactance Xi are
connected
together,
while running out of synchronism with each
other.
From these two equations fellows:
Maximum (effective) value of the fluctuating interchange current:
(20)
(21)
9
_
2x x+x 2+x
and, actual induced voltage:
where e = nominal induced voltage, effective value.
If the alternators are connected through an impedance z, z takes
the place of x, combining vectorially with xi and x2 .
In this calculation, the armature reaction has been assumed as
demagnetizing, and the impedance voltage therefore subtracted from
the nominal induced voltage.
This appears
correct,
as the inter-
change current between the
alternators out of synchronism with
each other, is essentially a lagging current, throughout, as illustrated
in Fig.
1.
If the two alternators are in synchronism, but out of phase with
each other by a maximum phase displacement angle 2co
, it is:
2E
sin
o> =I (2
[[END_PDF_PAGE:42]]

[[PDF_PAGE:43]]
Report of Charles P. Steinmetz
37
and, again assuming the armature reaction as demagnetizing:
v
Ixi
^0=60
n~
thus:
the maximum
(effective) value of the fluctuating exchange
current:
2xi-fx+Xj sin w
and, actual induced voltage:
E^XI+X
._..
o
eo
"~*
i.***/
where eo is the nominal induced voltage, effective value.
However, in this case of alternators in synchronism but oscillating
against each other, at least for small and moderate values of w
the
interchange current I
is essentially an energy current with regards
to the machine voltage, and the reactive component of this current
alternately changes between lag and lead, that is, between demagnet-
izing and magnetizing.
Therefore,
the correctness
is doubtful of
subtracting the impedance voltage from the nominal induced voltage
to get the induced voltage, but it would be:
E = Ve
! -i 2x 2
2
and as i varies between
and I, the average of E would be the mean
between e and
Ve
2
I*xj
J
, thus:
combining with the equation:
2Eo sin wo = I (2xx+x)
gives:
T _
2e
(2xi+x) sin w
~(2x,+x)
2-fx 3
2 sin 2w
E
(2xi+x)'
(2xi+x)
2 -|-x 2
2 sin 2w
It
is probable that the true value of E
lies between (23) and
(25), but nearer to (25).
Substituting these values
(21),
(23),
(25) into the equations of
A, B and C, and substituting
in these equations, as the impedance of the circuit between the two
alternators, gives the equations tabulated in F.
The
nominal
induced EMF
e
is
derived
by
combining the
terminal voltage e with the impedance voltage iz, where z is the total
[[END_PDF_PAGE:43]]

[[PDF_PAGE:44]]
38
Report of Charles P. Steinmetz
impedance inside of the terminals, true reactance as well as effective
reactance
of
armature
reaction.
For
non-inductive
load
and
synchronous machine load may be assumed as approximately non-
inductive
this gives:
= Ve
2 +(ix)
= e Vl+
2
(26)
where
is the percentage reactance, and the resistance is neglected,
as small compared with the reactance.
However, this expression neglects the change of reactance with
increase of magnetic saturation,
increase of magnetic leakage be-
tween field poles, etc., and therefore, especially hi turbo-alternators
with their enormous magnetic fields, high saturation and high field
leakage, this expression is not very accurate, and reasonably reliable
only in the mean range of current and voltage.
In B, and C, the case of two alternators, or groups of alternators
out of synchronism with each other, the equations of synchronizing
power, energy and critical slip :
p, p
, W, S
, contain the term
2xi+x
2xi+x+x 2
thus are a maximum, if this term is a maximum.
This is the case if:
x 2 = 2xi+x, or
x = x 2
2xi
that is:
The synchronizng power between alternators out of synchronism
with each other
is a maximum, and the frequency difference from
which they
pull each
other
into
synchronism,
is
greatest,
if the
alternators or group of alternators are connected together through a
reactance which
is
equal
to
the
effective
reactance
of armature
reaction, less twice the self inductive reactance of the circuit between
the alternators or groups of alternators.
With two alternators or
groups
of
alternators
connected
together
without
any
external
reactance, this means, if the self inductive reactance of the alternators
or group of alternators is one-third the synchronous reactance.
With
turbo-alternators, the self inductive reactance usually is much less,
and with such machines the synchronizing power thus
is increased
by the insertion of external reactance.
Substituting above relation into the equations of B, and C, gives
as the expressions for the case of maximum synchronizing power:
[[END_PDF_PAGE:44]]

[[PDF_PAGE:45]]
Report of Charles P. Steinmetz
39
Actual machine EMF:
Resultant EMF :
e = e
sin
en sin
i
Resultant current:
io =
Power fluctuation of low frequency :
p =
~
-
4X2
e
2
Energy transfer of low frequency :
W =
r>
*
f
T-*
Contmuous power transfer :
P =
Critical slip :
s =
a
cep
2 sin a
8 Xj
So
4 V27rfx 2M
[[END_PDF_PAGE:45]]

[[PDF_PAGE:46]]
40
Report of Charles P. Steinmetz
a
03
ffl
111
3
2 -s
"i
ts
x
o
.&
S
"3
It*
III
* ss
I
3 *
S
I
I
1w
o
.
Id
<u
C
ISu
(8
r-
u.
S
Id
* +
11
9
i
1
u
w
[[END_PDF_PAGE:46]]

[[PDF_PAGE:47]]
Report of Charles P. Steinmetz
41
[[END_PDF_PAGE:47]]

[[PDF_PAGE:48]]
42
Report of Charles P. Steinmetz
w
w
N
CO
a
3
cU
.&
co
S2
2
-x
V
60
2
+
w
x
>
W
[[END_PDF_PAGE:48]]

[[PDF_PAGE:49]]
Report of Charles P. Steinmetz
43
Denotations
e = nominal induced E. M. F. of alternator or group of alternators.
Xii = true
self inductive reactance of alternator or group of alter-
nators.
xn= external reactance of alternator or group of alternators, thus.
Xi = xn+Xi2 = total self inductive reactance of alternators or group
of alternators.
xj = effective reactance of armature reaction of alternator or group
of alternators, thus:
Xo=xn+x 2
synchronous
reactance
of
alternator
or
group
of
alternators.
x = reactance (or impedance) between alternators.
z = impedance of circuit between alternators.
=
Vi- 2+(2x!+x)
2
, where
r = resistance of circuit between alternators.
Or
approximately
a = phase angle of circuit between alternators, where:
tan
a =
Or approximately
:
a = 90 degrees.
w = phase displacement from mean, of oscillating alternators, thus:
2co = total phase displacement of oscillating alternators from each
other.
oj = maximum phase displacement during oscillation.
woo = initial value of w
pf= frequency of oscillation.
s = slip of frequency from mean, thus :
2s = slip of frequency of alternators from each other, and
sf= frequency of slip from synchronism;
2sf= frequency of slip of alternators from each other.
f= synchronous frequency.
to= time required for magnetizing effect of armature reaction.
M = momentum, in joules, of alternator or group of alternators, per
phase, that is:
3M = total momentum.
[[END_PDF_PAGE:49]]

[[PDF_PAGE:50]]
44
Report of Charles P. Steinmetz
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I
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6c
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II
fc
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CM
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in
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SO
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CM
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CO
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CM
CO
CM
in
r-l
CO
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CO
so
CO O
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I-H
so
CM *
r-l t-
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^ M
CM
in O O
ON
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CO
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in O
r-|
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M
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[[END_PDF_PAGE:50]]

[[PDF_PAGE:51]]
Report of Charles P. Steinmetz
45
H
In the trouble of September 18th, 1919, the following machines
were involved
:
In Fisk B and Northwest Station :
123,000 KW rating, 105,000 KW load = 85.5%
In Fisk A and Quarry Street:
114,000 KW rating,
90,000 KW load = 79%
It was:
Fisk A:
Six 12,000 KW; 98% reactance.
79% load.
Thus:
Vl+.79x.98
2 = 1.265
e =1.265
x
5200 = 6600V.
x 2 = 5.76;
Xl =.458 +.405 = 863;
M = 50xl0 6
.
Quarry St. :
Three 14,000 KW:
91% reactance.
79% load.
Thus:
Vl+.79x.91
2 =1.23.
e =6400V.
x 2 =4.60:
Xl = .318+.347 = .665; M = 67xlO.
FiskB:
Four 12,000 KW;
98% reactance.
85.5% load.
Thus:
Vr+.855
2 x98 2 = 1.305.
e = 6770V.
One 25,000 KW was immediately shut down.
Northwest Station:
One 20,000 KW:
134% reactance, and one 30,000 KW:
125%
reactance.
85.5% load.
20,000 KW:
Vl-f.855
2 x 1.34 2 =1.52.
e = 7900V.
x 2 = 4.95; Xl =.325+.168 = .493; M = 78xl0 6
.
30,000 KW:
Vl+ ,855 2 x 1.25 2 = 1.46:
e = 7600 V.
x 2 = 2.99:
Xl = .214+.168 = .382:
M = 87xl0 6
.
Busbar reactors :
x = 1.75
Tie cables
x =
.074
r = .312.
(1.)
Assuming,
at
first,
that
the two machines
in Northwest
Station are in sychronism with each other, and the four machines in
Fisk B are in synchronism with each other, but that Fisk B
is out
of synchronism with Northwest Station.
It
is then, in the circuit
between Fisk B and Northwest Station:
[[END_PDF_PAGE:51]]

[[PDF_PAGE:52]]
46
Report of Charles P. Steinmetz
Nominal induced EMF, average:
e = 7250 V.
Actual induced EMF:
E 2 =
.263 e = 1910 V.
Terminal voltage, average:
Et = .77E =1470 V.
EtV3 = 2550V.
Current between stations, maximum effective value:
io = 6400 A.
Max. pulsating power, per phase:
p = 6150KW.
Max. steady power transfer, per phase
:
p = 770KW.
Critical slip
:
so =.77%;
So
1 = .70%; So+So^l.47%
These values do not well agree with the observations recorded,
and while so many assumptions had to be made in the calculations
that the exact numerical values can not be relied upon too closely,
nevertheless,
the
general magnitude of the numerical values can
generally be relied upon.
It is probable that a terminal voltage of
2550 would not have escaped notice in the indicating meter ; a slowly
pulsating power transfer of 3p = 18,450 KW would have shown on
the record as a bad fluctuation, while the record shows an apparently
steady flow of about 1000 KW only.
Also, the fluctuating inter-
change current between the stations, rising to 6400 amperes, would
probably have shown marked distress in the cable.
It must therefore be assumed that the generators in the same
stations did not stay
in synchronism with each
other, and more
particularly that the 20,000 and the 30,000 KW unit in Northwest
Station broke out of synchronism.
The behaviour of these machines,
as the gradual loss of the field, is an indication in the same direction.
(2.)
With the 20,000 and the 30,000 KW units of Northwest
Station
out
of synchronism
with
each
other on
the busbars the
following relations would be obtained :
x, = 3.97; 2x! = .875 = z; M! = 78xl0 6
; M2 = 87xl0 6
.
e = 7750 V.
E =. 181 e = 1400 V.
io = 3200 A.
p = 2230KW.
So =.68%
so
1 =.64%
so+xo^l.32%
[[END_PDF_PAGE:52]]

[[PDF_PAGE:53]]
Report of Charles P. Steinmetz
47
Thus
with
the
breaking
of
the
synchronism
between
these
machines, the voltage and the circulating current materially dropped,
and with it also the current circulating between these machines and
Fisk B.
The 30,000 KW machine then was shut down, leaving the 20,000
KW as the only unit in Northwest Station, drafting out of synchro-
nism with Fisk B.
Assuming,
however, that
the four 12,000 KW
units in Fisk B have kept in synchronism with each other, the rela-
tions obtained:
(3.)
One 20,000 KW unit in Northwest Station; four 12,000 KW
units in Fisk B in synchronism with each other, but the two stations
out of synchronism with each other :
x 2 = 3.20:
2Xl =.709:
x=.074:
r = .312:
z = .84: 1^ = 78 xlO 6
:
M 2 = 200xl0 6
.
e =7330 V.
E =.208 e = 1520 V.
Et = .69 Eo = 1050 V.
Et V3 = 1820 V.
io = 3600 A.
p = 2750 KW.
so -.75%:
so
1 = .47%
80+80* = 1.22%.
While the values of p and
i
are much lower than in (1), still they
are much larger than indicated by the records; 3600 A. maximum
effective value of interchange current would give a loss in resistance
3 rin 2
of the tie cables of '=
= 600 KW,
in addition to the fluctuating
Zi
power reaching a maximum of 3p = 8250 KW.
Assuming then that not only the two stations, Fisk B and North-
west, but also the individual generators in either station had broken
synchronism.
The conditions then pertain:
(4.)
Two 12,000 KW alternators in Fisk B out of synchronism
with each other:
x 2 = 5.76:
2x! = 1.726 = z: M = 50xl0 6
.
e = 6770 V.
E =.23 63 = 1560 V.
Eo V3~= 2700V.
io=1820 A.
p=1410KW.
s =.67%
2s =1.34%
(5.)
The 20,000 KW
unit
in
Northwest
Station
against one
12,000 KW unit in Fisk Street, B.
[[END_PDF_PAGE:53]]

[[PDF_PAGE:54]]
48
Report of Charles P. Steinmetz
x 2 = 5.35: 2x!=1.356: x = .074:
r = .312:
z = 1.46: M^SOxlO
6
:
M 2 = 78xl0 6
.
e = 7330 V.
E =.215 e = 1570V.
Et = 61 Eo = 950 V.
Et = V3 = 1640 V.
io = 2150 A.
p = 1690KW.
s =.73%.
so
1 = .59%
so+so^l.32%
This means, with all the alternators in Fisk B and in Northwest
Station out of synchronism with each other, the terminal voltage
will be between
1640 and
nothing.
The interchange current cir-
culating between them will reach maximum amplitudes not exceeding
1820 to 2150 amperes effective.
The
i 2r loss in the tie cables then
would fluctuate between maximum values of 1500 to 2200 KW and
nothing, with a probable average of about 900 KW.
This agrees
with the observation that the wattmeter in the tie lines was very
steady showing about 1000 KW, and the voltage was in appreciable.
The conclusion therefore
is inevitable, that in this trouble, not
only
the Northwest
Station and
the
Fisk
Street B Station had
broken synchronism with each other, but that the individual gene-
ators in the Fisk Street B Station and in the Northwest Station had
broken synchronism with each other also, and were drifting past each
other out of synchronism.
The important question then
is, what caused these alternators
and stations to break synchronism.
That the Northwest Station and the Fisk Street B Station drifted
out of step with each other was to be expected.
As the
tie cable
which connect the bus bars of these two stations with each other
contain appreciable resistance but practically no reactance, and the
synchronizing power depends on the reactance,
there can be only
very
little synchronizing power between these stations, that
is, in
normal synchronous operations of these two stations,
it
is the in-
dividual generators of one station which synchronize with those of
the other station, rather than the stations as a whole.
However, the
alternators
in Fisk
Street B, have considerable
reactance, and no resistance between each other.
The reason for
their breaking synchronism with each other must be found
in the
great drop of voltage resulting from the break of synchronism be-
tween Fisk
Street B and Northwest
Station.
The
synchronizing
[[END_PDF_PAGE:54]]

[[PDF_PAGE:55]]
Report of Charles P. Steinmetz
49
power
is
proportional
to
the
square
of the
voltage. With Fisk
Street B and Northwest Station out of synchronism with each other
but the individual alternators in either station
still in synchronism
with each other (Case
1), the actual induced voltage in these two
stations drops
to an average of E =1910
volts per phase.
The
synchronizing power between two machines, for instance, two of the
12,000 KW alternators of Fisk Street B, reaches a maximum at a
phase displacement between two machines of 2co = 90 degrees, and
then is:
p =
= 2100 KW per phase,
or a total of 6300 KW, or about half load, and still less in (Case 3),
after the 30,000 KW unit had been shut down.
Thus,
if the load is suddenly released on these machines, unless
the steam supply can be reduced almost instantly to less than half
load, these machines will be torn out of synchronism with each other.
But by breaking synchronism between machines in the same stations
as the 20,000 and the 30,000 in Northwest Station
the voltage is
still further lowered and thereby the synchronizing power reduced,
so
that,
if one machine breaks out
of synchronism under
these
circumstances,
all
will break.
This must have happened on Sep-
tember
18th,
1919.
In
other
words,
the
break of synchronism
between Fisk Street B and Northwest Station lowers the voltage so
that there
is not
sufficient synchronizing power
left to keep the
individual machines in each station in synchronism with each other.
The important question then arises, what is theminimum machine
voltage necessary to assure the individual machines of each station
to remain in synchronism with each other irrespective of any sudden
release of full load.
That is, at what voltage does the synchronizing
power
of the individual machines become greater than
full
load.
This is gven by:
E
2
1
..
P=
grating
as shown in the following table:
TABLE
Lowest Voltage
at Which Machines Stay
in Synchronism,
If Full Load
is Thrown Off
Bating of Machine,
KW
12,000
14,000
20,000
25,000
30,000
35,000
Voltage per
Phase,
E =
2,630
2,490
2,560
2,740
2,770
2,650
Eo V"=
4,560
4,500
4,430
4,750
4,810
4,600
[[END_PDF_PAGE:55]]

[[PDF_PAGE:56]]
50
Report of Charles P. Steinmetz
As seen, as long as the actual induced voltage does not drop below
5000 volts there appears no danger of any of the machines breaking
out of synchronism with the other machines in the same station.
The insertion of sufficiently large power limiting reactors between
Fisk Street B and Northwest Station would in case of these stations
breaking synchronism with each other, maintain
sufficient voltage
in these stations, so that the machines in each station may be ex-
pected to keep in synchronism with each other.
Coming now to the consideration of the relation between Fisk
Street B and Quarry Street Station, during the trouble of September
18th, 1919:
(6.)
Four 12,000KW alternators in Fisk Street B, out of synch-
ronism over a power limiting reactor with three 14,000 KW alter-
nators in Quarry Street, the latter in synchronism with each other:
x 2 = 1.48:
2Xl = .438:
x=1.75:
z = 2.19:
Mi = 200 x 10 6
:
M 2 = 200xl0 6
.
e = 6585 V.
Eo = .597 e = 3930 V.
E
V3 = 6800 V.
= 3500 V.
Et V3 = 6000 V.
io = 3600 A.
p = 7000KW.
So = s
1 = .76%
s+s =1.52%
As seen, in Quarry Street, the voltage E
V3
is 6800, thus well
above the value required for stability of synchronous operation of
the alternators in this station, and no danger existed of their breaking
synchronism.
The record shows during the
disturbance, while Quarry Street
was connected with Fisk Street B out of synchronism, a fairly uniform
sustained
voltage
of 6800,
preceded by an
initial drop
of
short
duration, below 6000 volts.
The latter may be accounted for by the
feeding back from Quarry Street over the substations into Fisk B
and Northwest Station, which pulled the voltage of Quarry
Street
down, until the substations cut off, and required some time to recover.
The above calculated voltage of 6000
is lower than the
observed
voltage of 6800.
However, Quarry
Street was
also connected
to
Fisk Street A, and the latter station was assisting Quarry Street in
feeding the current over power limiting reactor B into Fisk B and
Northwest Station.
We may thus estimate the effect of Fisk A in
holding up the voltage.
[[END_PDF_PAGE:56]]

[[PDF_PAGE:57]]
Report of Charles P. Steinmetz
51
(7.)
Four 12,000 KW alternators in Fisk B, out of synchronism
over a power limiting
reactor with
three 14,000 KW alternators
in Quarry Street, the latter in synchronism with each other, and over
a second power limiting
reactor,
with
six 12,000 KW alternators
in Fisk A.
. x 2 =1.21:
2Xl = .414:
x=1.75:
z = 2.16: Mi = 200 x 10:
M 2 =500x 10 6
.
e = 6635 V.
Eo = .64 e = 4240 V.
E VT= 7350 V.
E t = 3800 V.
Et V3 = 6600 V.
io = 3900 A.
p = 8300KW.
so =-82%.
As seen, the calculated value of the terminal voltage of Quarry
Street, 6600, agrees as closely with the observed value of 6800 V. as
can
be
expected from
such approximated
calculations, especially
when
considering that some synchronous machines had been lost
by Quarry Street in the substations, and the load thereby reduced,
which would result hi an increase of voltage.
As the total impedance of Fisk Street A is about
1.1, and it
is
connected to Quarry Street by x=1.75, the voltage of Fisk Street A
should be higher than Quarry
Street
in
the proportion
of
1.1 to
1.1+ 1.75.
This gives for Fisk Street A: Et V3 = 8100 V.
This well
agrees with the average of the voltage record of Fisk Street A during
the first seven minutes of the disturbance.
Allowing for the continuously low voltage maintained at Fisk
Street B
by
the
out
of synchronism
condition with Northwest
Station,
the
fluctuating
current over
the
reactor B between
Fisk
Street B and Quarry Street would vary approximately between 1200
and 2700 amperes.
Assuming a temperature rise of these reactors
of 30C. at 600 A., this would give an estimated
final temperature
rise, at this excessive overload, of 600 to 800C., so that it should be
expected that after seven minutes, when
this reactor was discon-
nected, it would be very hot.
The reactor A, between Quarry Street and Fisk Street A, should
carry a current of 500
to
1000
amperes,
thus would not become
heated.
The conclusion then is
:
As the calculated numerical values agree
with the records of observation as closely as can be expected from
[[END_PDF_PAGE:57]]

[[PDF_PAGE:58]]
52
Report of Charles P. Steinmetz
such necessarily approximated calculations, it appears safe to accept
the correctness of the explanation that
Fisk Street A and Quarry Street remained in synchronism with
each other, but Fisk Street B and Northwest Station broke out of
synchronism
with
Quarry
Street
and
with
each
other, and the
individual machines of these latter two stations broke out of synchron-
ism with each other, and were
unable to pull into
step, due to
frequency
differences greater than
permissible by the small syn-
chronising power existing between these machines at the low voltage
existing due to the break of synchronism between the stations.
I
In the trouble of May
19th,
1919, the following machines were
involved :
Fisk A:
Five
12,000 KW = 60,000
KW.
Load
32,000 KW = 53%:
98% Reactance.
.53 2 x.98
2 = 1.13
e = 5900V.
x2 = 5.76:
Xl =.458+.405 = .863: M = 50xlO.
Quarry Street :
Four
14,000 KW = 56,000
KW.
Load
35,000 KW = 62.5%:
91% Reactance.
Vl+.625
2 x.91
2 = 1.15
e =6000V.
x 2 = 4.60:
Xl =.318+ .347 = .665: M = 67xlO.
FiskB:
Three
12,000 KW
and
one
25,000 KW = 61,000 KW. Load
45,000 KW = 74%.
98%:
>100% Reactance.
.74 2 x.98 2 =1.24
e = 6450V.
Vl+.74
2 x 1.20 2 =1.30
e = 6750V.
25,000, estimated:
x 2 = 4.0:
xi = .45: M = 85xlO.
Northwest :
Two 20,000 KW and one 30,000 KW = 70,000 KW.
Load50,OOOKW = 71%.
134%:
125% Reactance.
20,000:
Vl+ .71 2 x 1.34 2 =1.38
e = 7170V.
x 2 = 4.95:
xi =
.325+. 168 = .493; M = 78xl0 6
.
30,000:
Vl+ .71 2 x 1.25 2 = 1.34
e =7000 V.
x 2 = 2.99:
Xl = .214+.168 = .382 M = 87xl0 6
.
Busbar reactors :
x = 1.75
Tie cables:
x=
.074;
r=.312.
[[END_PDF_PAGE:58]]

[[PDF_PAGE:59]]
Report of Charles P. Steinmetz
53
Let us assume at
first that these stations have dropped out of
synchronism with each other, due to the trouble in Fisk A, and are
drifting past each other, and calculate by the preceding equations
for this condition the voltages and currents in the different stations,
and compare them with the recorded values, to see whether this
assumption is reasonable.
It is then:
(1.)
Fisk A out of synchronism with Quarry Street:
x 2 = (1.15+1.15)^-2 = 1.15;
2x 1 = .173+.166 =
.
x=1.75
z = 2.09
e = 5950V.
E =
?- e =.645e =3840 V.
z-f-x 2
Et =
Eo = 3530 V.
Et V3~= 6100 V.
(2.)
Quarry Street out of synchronism with Fisk B and Fisk A :
x 2 =(1.15+.61)-7-2 = .88;
2Xl = .166+.087 =
.
x = .875;
z = 1.128
E =.564e =
E=3120V.
Et V3 = 5400 V.
io=6250 A.
(3.)
Fisk B out of synchronism with Northwest Station:
x 2 = (1.30+ 1.36) -=-2 = 1.33; 2Xl = .176+.150 = .326; x = .074;
r = .312;
z = .507.
e =6830 V.
E =.276e =1880 V.
E=1530 V.
Et V3 = 2650 V.
io=7400 A.
For the reasons discussed before, it is not probable the individual
machines in Fisk B and Northwest Station would stay in synchronism
with each other.
As the relatively high terminal voltage recorded in
the stations shows that the individual machines stayed
in synchro-
nism with each other, the case thus may be considered of Fisk B and
Northwest
Station
in
synchronism with
each
other,
but
out
of
synchronism with Quarry Street:
[[END_PDF_PAGE:59]]

[[PDF_PAGE:60]]
54
Report of Charles P. Steinmetz
(4.)
Fisk B and Northwest Station in synchronism with each
other, but out of synchronism with Quarry Street :
X2 = (.683+ 1.15)4- 2 = .92; 2x! = .13+.166 = .299; x = 1.75;
z = 2.05.
eo- 6840V.
E =.69e = 4720'V.
Et =4250 V.
Et V3 = 7350V.
io = 4600 A.
Herefrom then we
get the
terminal voltages
of the
different
stations
:
Fisk A:
Quarry:
Fisk B:
Nw. St.:
(1.)
All 4 stations out
of synchronism :
Calculated
ter-
minal voltage.
.
.
6100
5400
2650
2650
Observed
records
7800 to 8200
7900 to 8100
8300 to 8400
8400 to 8500
8100 to 8500
(2.) Fisk B and Nw.
Station in synchronism:
Calculated
6100
5400
7350
7350
Observed
records
7800 to 8200
7900 to 8100
8300 to 8400
8400 to 8500
8100 to 8500
As seen, the calculated values are uniformly very much lower
than the observed values, and while no very great accuracy can be
attributed to such approximated calculations, the difference
is too
great to be accounted for, and it must therefore be concluded that in
this trouble none of the stations had broken synchronism, but
all
the four stations had kept in synchronism with each other.
Assuming then as reasonable, that in the trouble of May 19th,
the
four
station
sections had kept
in synchronism,
the question
remains to account for the great drop and fluctuations of voltage,
which were greatest at the source of the trouble, Fisk Street A, and
decreased towards the other end of the station chain, whether due
to the hunting of the stations against each other, or due to excessive
load of lagging current, caused by starting of synchronous machines,
or due to some other cause.
Assuming
first, that the voltage drop and voltage fluctuations
was due
to
the
simultaneous
starting
of numerous
synchronous
machines.
To estimate the effect thereof, in the following table are
given, in columns (1) to (3), the recorded voltages of the four stations:
[[END_PDF_PAGE:60]]

[[PDF_PAGE:61]]
Report of Charles P. Steinmetz
55
The initial minimum and the final minimum towards the end of the
disturbance, and
the average
fluctuation.
In
the second
line
is
given the average phase voltage, that is terminal voltage divided by
V3.
Column (4) then gives the average drop of voltage; column (5)
the
total
station impedance,
true
reactance
as
well
as
effective
reactance of armature
reaction.
As we have to do here with
a
sustained
voltage
drop,
the armature
reaction comes
into
play.
Column
(6) then gives the average value of the lagging current in
each station, which would account for this voltage drop.
As seen
these are fairly moderate currents.
This would be the lagging current
drawn from the stations in starting the synchronous machines in the
substations.
As (practically)
all the synchronous machines in Fisk
A had dropped out, but only a few on the other stations, most of this
lagging load would be on Fisk Street A.
Assuming then that half
this current had been consumed directly from the stations, the other
half transferred over the power limiting reactors to Fisk Street A.
Column (7) would then give the average lagging current consumed in
each station, and Column (8) the current transferred over the power
limiting
reactors between
the
stations,
to
the
next
station, and
Column (9) gives the load on each station, in KYA.
As Fisk Street A
lost a load of about 32,000 KW in synchronous machines, a load of
12,000 KVA starting current of these synchronous machines, during
eighteen minutes, does not appear excessive, and the
lesser values
in the other
stations
also appear of reasonable magnitude.
The
voltage across the power limiting reactors, in Column (10). and the
phase angles between the stations, in Column (11), are very moderate:
only a
little over 5 degrees maximum phase displacement between
Fisk A and Quarry
Street.
That
is, the strain
is very moderate,
and very far from the limits of synchronizing power.
Assuming now the second explanation, hunting of the stations.
Column
(12)
then
gives the
estimated
values
of the quadrature
voltage resulting from the swing of the machines, which would be
required to bring about the observed voltage drop:
V5200 2
e
2
where 5200
is the normal phase voltage of the stations, and e the
observed phase
voltage.
The
total phase angle of swing then
is
given as 2 a? 1 in Column (13).
As the limits of synchronizing power
are at 90 degrees phase angle, and the maximum estimated phase
angle of swing is 30 degrees,
it would appear
if this explanation is
correct
that
the
stations were
still
far from the
limits
of
their
synchronizing power, that is, quite stable.
Column (14) then gives
the true self inductive reactance of the stations, and Column
(15)
[[END_PDF_PAGE:61]]

[[PDF_PAGE:62]]
56
Report of Charles P. Steinmetz
gives two values of the surging current in the stations resulting from
the oscillations of the machines.
The lower value is calculated by
the total station impedance, Column
(5), the higher value by the
self inductive impedance Column
(14).
The true value would be
intermediate, probably nearer the
larger
figure,
as the period
of
oscillation is too fast to develop more than a small part of the effective
reactance of armature reaction.
Either of the two assumptions gives values of magnitude which
are reasonable and in accordance with the observed data, so that in
the absence of any data on the phase angle between the stations, it
is not possible to decide which is the correct explanation.
It must
be
recognized,
however,
that
neither
of the two
explanations
is
entirely satisfactory, as either
fails to account sufficiently for the
excessive heating of the reactor B between Quarry Street and Fisk
Street B.
This makes it the more desirable, in view of the import-
ance of fully understanding the phenomenon, to install phase angle
indicators, that is suitable synchronoscopes, in all the stations, and
carefully observe them in case of any trouble in the system.
[[END_PDF_PAGE:62]]

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Report of Charles P. Steinmetz
57
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Report of Charles P. Steinmetz
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RARE
6K
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