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```FIGURE 4.35 Voltage buildup
in a self-excited de generator.
r·
154 chapter 4 DC Machines
Critical field circuit resistance.
FIGURE 4.36 Effect of field resis-K:....---------------1, tance.
terminal voltage is V, = InRr < Eal· The increased armature voltage Ea1 will
eventually increase the field current to the value lr2 , which in turn will build
up the armature voltage to Ea2 • This process of voltage buildup continues.
If the voltage drop across Ra is neglected (i.e., Ra ~ Rr), the voltage builds
up to the value given by the crossing point (Pin Fig. 4.35) of the magnetiza-
tion curve and the field resistance line. At this point, Ea = IrRr = V, (assume
Ra is neglected), and no excess voltage is available to further increase the
field current. In the actual case, the changes in Ir and Ea take place simultane-
ously and the voltage buildup follows approximately the magnetization
curve, instead of climbing the flight of stairs.
Figure 4.36 shows the voltage buildup in the self-excited de generator for
various field circuit resistances. At some resistance value Rf3, the resistance
line is almost coincident with the linear portion of the magnetization curve.
This coincidence condition results in an unstable voltage situation. This
resistance is known as the critical field circuit resistance. If the resistance is
greater than this value, such as Rr4 , buildup (V,4 ) will be insignificant. On
the other hand, if the resistance is smaller than this value, such as Rfl or
Rr2 , the generator will build up higher voltages (Vtl, V,2). To sum up, three
conditions are to be satisfied for voltage buildup in a self-excited de genera-
tor:
1. Residual magnetism must be present in the magnetic system.
2. Field winding mmf should aid the residual magnetism.
3. Field circuit resistance should be less than the critical field circuit resis-
tance.
EXAMPLE 4.3
The de machine in Example 4.2 is operated as a self-excited (shunt) generator
at no load.
f
T DC Generators 15 5
(a) Determine the maximum value of the generated voltage.
(b) Determine the value of the field circuit control resistance (Rrc) required
to generate rated terminal voltage.
(c) Determine the value of the critical field circuit resistance.
Solution
(a) The maximum voltage will be generated at the lowest value of the field
circuit resistance, Rrc = 0. Draw a field resistance line (Fig. E4.3b) for
Rr = Rrw = 80 !1. The maximum generated voltage is
(b)
lllV
100 100 v
80
.l!l
~ 60
fol;,
40
20
FIGURE E4.3
Ea = 111 volts
Vt = Ea- laRa
It
=Ea
= 100V
Ia
(a)
11, amps
(b)
I
I
I
I
I
I
I
I
I
I
+
R1 = 80 !l
156 chapter 4 DC Machines
Draw a field resistance line that intersects the magnetization curve at
100 V (Fig. E4.3b ). For this case,
Ir = 1 A
100 R1 = - 1- = 100 D = Rrw + Rrc
Rrc = 100- 80 = 20 D
(c) Draw the critical field resistance line passing through the linear portion
of the magnetization curve (Fig. E4.3b). For Ir = 0.5, Ea is 85 V.
85
Rrccritl = 0.5 = 170 D
Rrc = 1 70 - 80 = 90 D •
Voltage-Current Characteristics
The circuit of the self-excited de generator on load is shown in Fig. 4.37.
The equations that describe the steady-state operation on load are
Ea = Vt +laRa
Ea = Ka<Pwm = function of Ir
( 4.25a)
(4.25b)
~magnetization curve (or open-circuit saturation curve)
Vt = IrRr = Ir(Rtw + Rrc) ( 4.25c)
(4.25d)
( 4.25e)
Vt =hRL
fa= /1 + h
The terminal voltage (Vt) will change as the load draws current from the
machine. This change in the terminal voltage with current (also known as
voltage regulation) is due to the internal voltage drop laRa (Eq. 4.25a) and
the change in the generated voltage caused by armature reaction (Eq. 4.25b).
In finding the voltage-current characteristics (Vt versus Ia) we shall first
neglect the armature reaction and then subsequently consider its effects.
+
FIGURE 4.37 Self-excited de generator
with load.
?
DC Generators 15 7
Without Armature Reaction
The voltage-current characteristic of the self-excited generator can be ob-
tained from the magnetization curve and the field resistance line, as illus-
trated in Fig. 4.38. Note that the vertical distance between the magnetization
curve and the field resistance line represents the I.R. voltage drop. Consider
the various points on the field resistance line, which also represents the
terminal voltage V 1. For each terminal voltage, such as V11, compute the
armature current I.1 from the I.R. voltage drop, which is the vertical distance
between V 11 and £.1 • If this calculation is performed for various terminal
voltages, the voltage-current characteristic of the de generator, shown in
Fig. 4.38b, is obtained. Note that (Fig. 4.37) at I1 = 0, I. = Ir, and therefore
the actual no-load voltage, V10 , is not the voltage given by the crossing point
P of the magnetization curve and the field resistance line, as predicted earlier
because of neglecting R •. However, for all practical purposes V1a = VP.
A convenient way to construct the voltage-current characteristic from the
magnetization curve and field resistance line is to draw a vertical line at
point P. This vertical line represents the I.R. drop. In Fig. 4.38, the vertical
r
% rated field current
(a)
I
I
I
I
I
I
I
I
-------r-------
1
I
I
I
I
(b)
1a(max)
FIGURE 4.38 Terminal characteristic of a self-excited de generator.
(
r
..
I" ~,.
158 chapter 4 DC Machines
line pq represents the voltage drop laRa. A line qbn is drawn parallel to Op.
Therefore pq = ab = mn = la1Ra. The same armature current results in two
terminal voltages, Vtt and Vt2 • To obtain the value of the maximum armature
current that can be drawn from the de generator, a line rs is drawn parallel
to Op and tangential to the magnetization curve. This will result in the
maximum vertical distance, sk, between the field resistance line and the
magnetization curve. Also note that if the machine terminals are shorted
(i.e., RL ___,. 0), the field current is zero and the machine currents Ua = It =
h = Eaf Ra) are not very high. However, before RL is reduced to zero, the
armature current may be large enough (such as the current Ia(max) in Fig.
4.38b) to cause damage to the machine.
From Figs. 4.29 and 4.38b it is apparent that the terminal voltage drops
faster with the armature current in the self-excited generator. The reason
is that, as the terminal voltage decreases with load in the self-excited genera-
tor, the field current also decreases, resulting in less generated voltage,
whereas in the separately excited generator the field current and hence the
generated voltage remain unaffected.
With Armature Reaction
When armature current flows it produces an internal voltage drop laRa. If
the armature produces demagnetizing effects on the pole, there will be a
further voltage drop in the terminal voltage. The terminal voltage will there-
fore drop faster than shown in Fig. 4.38b in the presence of armature reac-
tion.
In Fig. 4.39, let pq ( = laRa) represent the voltage drop for a particular
value of the armature current Ia. If armature reaction is not present, the
terminal voltage is Vtt. Let qr ( = Ir(AR)) represent armature reaction in equiva-
11, amps
FIGURE 4.39 Determination of terminal voltage.
'T'
I
DC Generators 15 9
lent field current for this value of armature current. A line rc is drawn parallel
to Op and intersects the magnetization curve at c. The triangle pqr is drawn
as abc such that a is on the field resistance line and c is on the magnetization
curve. Therefore, in the presence of armature reaction, the terminal voltage
is V,., which is lower than V, 1 , where Vtt is the terminal voltage if armature
reaction is not present. Note that V, = V,., E. = V, + I.R. = V,. + ab and
If(efO = Ir - Ir(AR) = Ir - be. The terminal voltage corresponding to any other
value of the armature current can be determined by constructing```