IEEE Solid-States Circuits Magazine - Spring 2018 - 10
Linear Circuit:
Resistors and
Sources
+
-
+
-
iL(t )
+ vout(t )
-
+
Linear Circuit:
Resistors and
Sources
RL vout(t )
-
(a)
RL
voc(t ) and Req
Linear Circuit:
Resistors and
Sources
+
-
isc(t )
Original Circuit with
vout(t )/Req
Indep. Sources
Zeroed
+
-
(a)
+ vout(t )
-
(c)
Figure 5: An intuitive proof of the Norton
theorem. (a) The load is replaced by a
voltage source having a voltage identical
to the load. The output current is a result
of contributions from all the independent
sources shown in red. (b) The contribution
of independent sources inside the box to
the output current is determined by zeroing
the output voltage. (c) The contribution of
the output voltage is determined by zeroing
all the independent sources inside the box.
same equation. This completes our intuitive proof of the Thévenin theorem.
The proof for the Norton theorem
is similar. Here, we take the output
current as the output variable and
replace the load with an ideal voltage
source v out (t) as shown in Figure 5(a).
Using superposition, as illustrated in
Figure 5(b) and (c), we can write
i L (t) = i sc (t) - v out (t) /R eq .
Since both Thévenin and Norton
equivalent circuits represent the same
circuit, they are equivalent themselves. Combining the two equations
above, we can see that
R eq = voc (t) /i sc (t) .
inside the black box and the independent current source representing
the load. To find the output voltage
due to the sources inside the box, we
will zero the current source i L (t) as
shown in Figure 4(b). This is equivalent
to saying we open the load and find the
voltage. We already called this voltage
v oc (t). Next, to determine the voltage
due to the current source i L (t), we zero
all the independent sources inside the
box, as shown in Figure 4(c). This will
result in a simple equivalent resistor
R eq with - i L (t) as its current. Therefore, the total output voltage can be
written as
v out (t) = v oc (t) - i L (t) R eq .
This same output voltage can
be obtained when the black box is
replaced with a voltage source v oc (t)
in series with a resistor R eq [see Figure 2(c)], as both are governed by the
s p r i n g 2 0 18
(b)
vout(t )
-
vout(t ) and ReqiRL
Figure 6: (a) The resistive load is separated from the rest of the circuit, and the open-circuit
voltage is distinct from the output voltage. (b) The resistive load is lumped with the rest of the
circuit, and the new open-circuit voltage is equal to the output voltage. The new equivalent
resistor is a parallel combination of the original equivalent resistance and the load resistance.
+
-
(b)
10
+
Linear Circuit:
Resistors and
Sources
We end this article by recognizing two special cases of Thévenin
theorem.
■ A case where the load is an open
circuit, i.e., where the load draws
zero current from the output node.
In this case, the output voltage is
equal to v oc (t), simply because
there is no contribution to the output voltage from the load. Yet we
must include R eq in series with this
voltage source to fully model this
circuit because we wish our model
to be valid even if we decide to
change the load. In other words, the
model should be independent of
the load and should work no matter
the load.
■ A case where the load is a linear
resistor, say R L, as shown in Figure 6(a). In this case, we can push
the load into the black box or,
equivalently, draw a bigger box that
IEEE SOLID-STATE CIRCUITS MAGAZINE
includes the original box and the
load, as shown in Figure 6(b). By doing so, the new v oc (t) (corresponding to the bigger box) will be equal
to the output voltage; yet, similar to
the previous case, we must include
the equivalent resistance in series
with this voltage source. The equivalent resistance in this case is the
parallel combination of the original
equivalent resistance (which we
called R eq) and R L.
The Thévenin/Norton theorems
explain why two nodes that have the
same voltage may not be identical: because they may have different equivalent resistances in series. For example,
we may measure 0.5 V at two different
nodes in a circuit, but one may reduce
to 0.1 V if we connect it to a 1-kX resistor, whereas the other may stay
close to 0.5 V even after we connect it
to a 1-kX resistor. In the former case,
the equivalent resistance of the node
has been 4 kX, whereas in the latter,
the equivalent resistance has been
around zero!
In summary, we have provided an
intuitive understanding and proof
for each of the Thévenin and Norton
equivalent circuit theorems. In the
next issue, we will provide several
circuit examples where Thévenin and
Norton equivalent circuits offer significant insight into circuit analysis
and design.
Reference
[1] R. E. Thomas, A. J. Rosa, and G. J. Toussaint, The Analysis and Design of Linear
Circuits, 7th ed. New York: Wiley, 2012.
Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Spring 2018
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