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258 ⏐⏐⏐ METHODS OF ANALYSIS AND SELECTED TOPICS (dc)
R s
E
E
I =
Rs R L
R L
I L
FIG. 8.6
Practical voltage source.
R s
FIG. 8.7
Practical current source.
I L
sources—whether they are voltage or current—have some internal resistance
in the relative positions shown in Figs. 8.6 and 8.7. For the voltage
source, if Rs � 0 � or is so small compared to any series resistor that it
can be ignored, then we have an “ideal” voltage source. For the current
source, if Rs � ∞ � or is large enough compared to other parallel elements
that it can be ignored, then we have an “ideal” current source.
If the internal resistance is included with either source, then that
source can be converted to the other type using the procedure to be
described in this section. Since it is often advantageous to make such a
maneuver, this entire section is devoted to being sure that the steps are
understood. It is important to realize, however, as we proceed through
this section, that
source conversions are equivalent only at their external terminals.
The internal characteristics of each are quite different.
We want the equivalence to ensure that the applied load of Figs. 8.6
and 8.7 will receive the same current, voltage, and power from each
source and in effect not know, or care, which source is present.
In Fig. 8.6 if we solve for the load current IL, we obtain
E
IL � �� (8.1)
Rs � RL If we multiply this by a factor of 1, which we can choose to be Rs/Rs, we obtain
(1)E
IL � �� � �
Rs � RL ( Rs/Rs)E
� � �
Rs
� RL
Rs(
E/Rs)
R
� � �
sI
� (8.2)
Rs
� RL
Rs � RL If we define I � E/Rs, Equation (8.2) is the same as that obtained by
applying the current divider rule to the network of Fig. 8.7. The result
is an equivalence between the networks of Figs. 8.6 and 8.7 that simply
requires that I � E/Rs and the series resistor Rs of Fig. 8.6 be placed in
parallel, as in Fig. 8.7. The validity of this is demonstrated in Example
8.4 of this section.
For clarity, the equivalent sources, as far as terminals a and b are concerned,
are repeated in Fig. 8.8 with the equations for converting in
either direction. Note, as just indicated, that the resistor Rs is the same in
each source; only its position changes. The current of the current source
or the voltage of the voltage source is determined using Ohm’s law and
the parameters of the other configuration. It was pointed out in some
detail in Chapter 6 that every source of voltage has some internal series
resistance. Forthe current source, some internal parallel resistance will
always exist in the practical world. However, in many cases, it is an
R s
E = IR s
a
b
FIG. 8.8
Source conversion.
I = E R s
R s
a
b
N A