Ideal Energy Sources:
Voltage Source:
In reality, all voltage sources (e.g., a battery or a
voltage amplifier) have a nonzero internal resistance
in series with
the voltage source that causes the actual output voltage
to be lower
than
, depending on the load current
. The load voltage
and the
load current
are constrained by the following two relationships imposed
by the voltage source
and the load
, respectively:
The slope of the first curve is the internal resistance
and the slope
of the second curve is
. Solving these two equations we get load voltage
and current
.
Only in the case of an ideal voltage source with
will
.
For
, the heavier the load, i.e., the smaller
, the larger the
load current
, and the lower the load voltage
:
Current Source:
In reality, all current sources (e.g., a solar cell
or a current amplifier) have an internal resistance
in parallel
to the current source that causes the actual output current
to be lower
than
, depending on the load voltage
. The load voltage
and the
load current
are constrained by the following two relationships imposed
by the current source
and the load
, respectively:
The slope of the first curve is the internal resistance
and the slope
of the second curve is
. Solving these two equations we get load voltage
and current
.
Only in the case of an ideal current source will
and
.
For
, the larger the load resistance
, the smaller the
current
.
Equivalent Circuits and Source Transformation:
Two circuits with the same voltage-current relation (V-I characteristics) at the output port are considered equivalent, as they have the same external behavior, although they may be different internally.
A voltage source can be equivalently converted into a current source and vise versa:
Finding internal resistance:
As shown above, the internal resistance
of voltage source is the slope of
the straight line
, and similarly, the internal resistance
of
the current source is the slope of the straight line
. In both cases
this slope can be obtained as the ratio of the open-circuit voltage (when
) and the short-circuit current (when
):
Using this method, one can determine the internal resistance
of a
given energy source theoretically. However, it may not be practical to do so
experimentally, as the short circuit current is difficult to obtain without
causing damage of the source (a battery, or a signal generator). In this case,
we can find the voltage
and current
(
) associated with each
of two load resistors
and
. The internal resistance
can then
be found as the slope of the straight line determined by the two points
and
.
Example 1:
A voltage source of
and
can be
converted to a current source of
with the same
(and vice versa). For a load of
, both energy sources will
provide the load current
and load voltage
. We see
that this is a good voltage source but a poor current source, as its
is too low.
Example 2:
A current source of
and
can
be converted to a voltage source of
with the same
(and vice versa). For a load of
, both energy sources will
provide the load current
and load voltage
. We see
that this is a good current source but a poor voltage source, as its
is too high.
Power Delivery/Absorption
Example 3:
In the circuit shown, the ideal current source is
,
and the ideal voltage source is
, the resistor is
.
Find the current through and voltage across each of the three components.
Find the power delivered, absorbed, or dissipated by each of the three
components.
The current source provides
current through the left branch
(upward), while the voltage source provides
across all three
components. The current through
is
(downward),
the current through the voltage source is
. Therefore
(Homework) Redo the above with the polarity of
reversed. Find:
Comment: While various voltage sources such as batteries are very
common in everyday life, current sources do not seem to be widely available.
One type of current source is solar-cell, which generates current proportional
to the intensity of the incoming light. Also, certain transistor circuits
are designed to output constant current. Moreover, as discussed above, any
voltage source can be converted into a current source. For example, a
current source with
mA and
can be implemented
by a voltage source of
in series with
.