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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