For any circuit containing
resistors and sources, the v-i relation will be of the form
v=
R
eq
i+
v
eq
v
R
eq
i
v
eq
(1)
and the
Thévenin equivalent circuit for any
such circuit is that of
Figure 1. This equivalence applies no matter how many
sources or resistors may be present in the circuit. In an
example, we know
the circuit's construction and element values, and derive the
equivalent source and resistance. Because Thévenin's
theorem applies in general, we should be able to make
measurements or calculations
only from the terminals
to determine the equivalent circuit.
To be more specific, consider the equivalent
circuit of Figure 1.
Let the terminals be open-circuited, which has the effect of
setting the current
i i to zero. Because no current flows
through the resistor, the voltage across it is zero (remember,
Ohm's Law says that
v=Ri
v
R
i
). Consequently, by applying KVL we have that the
so-called open-circuit voltage
v
oc
v
oc
equals the Thévenin equivalent voltage. Now
consider the situation when we set the terminal voltage to zero
(short-circuit it) and measure the resulting current. Referring
to the equivalent circuit, the source voltage now appears
entirely across the resistor, leaving the short-circuit current
to be
i
sq
=-
v
eq
R
eq
i
sq
v
eq
R
eq
.
From this property, we can determine the equivalent resistance.
v
eq
=
v
oc
v
eq
v
oc
(2)
R
eq
=-
v
oc
i
sc
R
eq
v
oc
i
sc
(3)
Use the open/short-circuit approach to derive the
Thévenin equivalent of the circuit shown in Figure 2.
v
oc
=
R
2
R
1
+
R
2
v
in
v
oc
R
2
R
1
R
2
v
in
and
i
sc
=-
v
in
R
1
i
sc
v
in
R
1
(resistor
R
2
R
2
is shorted out in
this case). Thus,
v
eq
=
R
2
R
1
+
R
2
v
in
v
eq
R
2
R
1
R
2
v
in
and
R
eq
=
R
1
R
2
R
1
+
R
2
R
eq
R
1
R
2
R
1
R
2
.
For the
depicted circuit, let's derive its Thévenin equivalent
two different ways. Starting with the open/short-circuit
approach, let's first find the open-circuit voltage
v
oc
v
oc
. We have a current divider relationship as
R
1
R
1
is in parallel with the series combination of
R
2
R
2
and
R
3
R
3
.
Thus,
v
oc
=
i
in
R
3
R
1
R
1
+
R
2
+
R
3
v
oc
i
in
R
3
R
1
R
1
R
2
R
3
.
When we short-circuit the terminals, no voltage appears across
R
3
R
3
, and thus no current flows through it. In short,
R
3
R
3
does not affect the short-circuit current, and can be eliminated. We
again have a current divider relationship:
i
sc
=-
i
in
R
1
R
1
+
R
2
i
sc
i
in
R
1
R
1
R
2
. Thus, the Thévenin equivalent
resistance is
R
3
R
1
+
R
2
R
1
+
R
2
+
R
3
R
3
R
1
R
2
R
1
R
2
R
3
.
To verify, let's find the equivalent resistance
by reaching inside the circuit and setting the current source
to zero. Because the current is now zero, we can replace the
current source by an open circuit. From the viewpoint of the
terminals, resistor
R
3
R
3
is now in parallel with the
series combination of
R
1
R
1
and
R
2
R
2
.
Thus,
R
eq
=
R
3
∥
R
1
+
R
2
R
eq
∥
R
3
R
1
R
2
,
and we obtain the same result.
