IEEE Solid-States Circuits Magazine - Fall 2023 - 12

constrained in any way by the elements
used in the original circuit.
For example, whereas the independent
voltage source in Figure 1(a)
is a 2-V step function, we have arbitrarily
assigned to its corresponding
branch a dc voltage of 4 V and
a dc current of −5 mA. Similarly,
even though the voltage across the
capacitor should be a decaying exponential
with a time constant of 1 µs,
we have assigned to this branch a
dc voltage of 3 V and a dc current
of 5 mA. The reader can verify, however,
that the assigned voltages and
currents do satisfy the KVL and KCL,
respectively. For example, we have
assigned ii
34
21
vv
=- to satisfy KCL and
= in accordance with the KVL.
What Tellegen's theorem claims is
that, for these assigned voltages and
currents, the sum of vi
kk over all k
is zero. This is indeed true for our
example. To demonstrate this further,
we have assigned a second set
of voltages and currents (vk
t
and )ik
t
to this graph, as shown in the bottom
two rows of Table 1. Again, the
sum of vi
t kk
t over all k is zero.
A special case would be to assign
the voltages and currents of the
actual circuit to the branches of its
graph. Clearly, in this case, because
the set of voltages and currents of
the original circuit satisfy both
the KVL and KCL, they form a valid
assignment, and Tellegen's theorem
applies. The result in this special
case should not be surprising
because
vt ,it
kk
^^
hh with the associated
reference directions we defined,
represents the instantaneous power
delivered to the element in branch k
at time t, and so the sum of all the
instantaneous powers must be zero
to satisfy the conservation of energy
principle. In other words, the energy
per unit time that is generated in the
circuit (say, by the independent voltage
source in Figure 1) must be equal
to the energy per unit time that is
consumed or stored by the rest of
the circuit. What may be surprising,
however, is that Tellegen's theorem
is valid for any independent set of
voltages and currents assigned to
the graph branches, as described
previously. How can you make sense
of this? The reader is encouraged
to pause and ponder this question
before reading further.
If we accept that the conservation
of energy principle holds
for any circuit, then with a set of
branch voltages and currents, such
as the ones given in the top two
rows of Table 1 and the graph of Figure
1(b), we can envision a circuit
with the same graph but with elements
whose ivcharacteristics
600
Ω
1 kΩ
250 Ω
4 V
(a)
1 V
+-
2 A 6 A
1,125 Ω
(b)
FIGURE 2: The two circuit realizations
corresponding to the graph of Figure 1(b)
that satisfy the voltage and current values
shown in (a) the top two rows of Table 1
and (b) the bottom two rows of Table 1.
12
FALL 2023
follow the data in Table 1. In other
words, we can always build a circuit
whose elements have the voltages
and currents that we specify. For
example, we can construct the circuit
of Figure 2(a), corresponding to
the top two rows in Table 1. Indeed,
if we try to find the branch currents
and voltages for this circuit, we will
arrive at the voltages and currents
shown in the top two rows of Table 1.
And, as this is a real circuit, we
expect the sum of powers delivered
to the elements to be zero. Similarly,
Figure 2(b) shows a circuit
realization corresponding to the
bottom two rows of Table 1. Again,
the reader can easily verify that the
solution to this circuit is consistent
with the numbers provided in Table 1.
What if we mix the voltages of row 1
IEEE SOLID-STATE CIRCUITS MAGAZINE
()vk
with the currents of row 4 ()?ik
t
The reader is asked to construct a
circuit corresponding to these
two rows.
To recap, we used the conservation
of energy principle to provide
an intuitive proof of Tellegen's theorem.
Here are three important conclusions
of Tellegen's theorem:
1) The conservation of energy in
circuits is a direct consequence
of the KVL and the KCL alone and
has nothing to do with the actual
elements in the circuit.
2) Modifying a lumped circuit by
replacing one or more of its elements
with other elements (such
as replacing a resistor with a
diode or a capacitor) will most
likely change the branch voltages
and currents of the circuit, say,
from vk
and ik
to vk
t
and
i ,k
t
respectively.
However, in doing so,
the following sums will remain
equal to zero:
////t tt tkk
k1111
b
====
====
vi
kk
k
vi vi
k
kk
kk
k
3) In mathematical terms, corresponding
to the b branches of
a graph, if we form a vector
of voltages and currents, denoted
by
v = (, ,, )
f
vv vvb12
ii iib12
and
v = f (, ,, ), respectively, then
the two vectors are orthogonal.
That is, their dot product is always
zero: ..
v
vi 0=
v
Finally, we note that the theorem
assumes lumped elements in its
statement simply because a branch
current cannot be defined for a
distributed, nonlumped element.
Indeed, the current through a distributed
element is not unique and
depends on the physical position in
the element.
In summary, Tellegen's theorem
states that the KVL and the KCL alone,
and not the circuit element characteristics,
imply the orthogonality of
the voltage and current vectors and
the conservation of energy principle.
In this article, we used the conservation
of energy principle to provide an
(continued on p. 84)
b
b
b
vi 0.
+
-
IEEE Solid-States Circuits Magazine - Fall 2023

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