IEEE Solid-States Circuits Magazine - Fall 2023 - 11

CIRCUIT INTUITIONS
Ali Sheikholeslami
W
" Tellegen's Theorem "
Welcome to the 37th article in the " Circuit
Intuitions " column series. As the
title suggests, each article provides
insights and intuitions into circuit
design and analysis. These articles are
aimed at undergraduate students but
may serve the interests of other readers
as well. If you read this article, I
would appreciate your comments and
feedback as well as your requests and
suggestions for future articles in this
series. Please e-mail me your comments
at ali@ece.utoronto.ca.
The field of circuit theory is
filled with elegant concepts, principles,
and theorems, but a handful
are most used in everyday circuit
designs. Of those, we can refer to
superposition, substitution, Thevenin's
and Norton's equivalent circuit
theorems, Tellegen's theorem,
and the reciprocity theorem. These
concepts have been introduced and
discussed at length in most of the
circuit textbooks, including [1]. In
the previous articles in this series,
we have discussed superposition
and Thevenin's and Norton's equivalent
circuit theorems [2], [3], [4]. In
this article, without providing an
elaborate proof, we explain Tellegen's
theorem and offer intuitions as
to why it makes sense.
Tellegen's Theorem
Consider a circuit with lumped elements
whose graph consists of n
nodes and b branches, and assign to
each branch k an arbitrary voltage
Digital Object Identifier 10.1109/MSSC.2023.3315668
Date of current version: 14 November 2023
vt ,
k^h subject to Kirchhoff's voltage
law (KVL) and an arbitrary current
itk^h in an associated reference
direction, subject to Kirchhoff's current
law (KCL), then
b
/vt it 0
k=1
kk^ h =
^ h
of vt it
kk
That is, the sum over all branches
^^
hhis zero at any time t.
Let us first review this theorem
2 kΩ
2 kΩ
1 nF
2u(t)
v2
+
i1
i2
v1
(a)
-
+
v3
-
(b)
FIGURE 1: (a) An example circuit with
three nodes and four elements and (b) the
corresponding graph with three nodes and
four branches.
+
i3 i4 v4
-
through an example, and then try to
make sense of it. Figure 1(a) shows
an example circuit that consists of
four elements: an independent voltage
source, two resistors, and one
capacitor. A corresponding graph for
this circuit, shown in Figure 1(b), has
a total of three nodes, as in the original
circuit, and four branches, corresponding
to the four elements in
the original circuit. To each branch,
say, branch k, we assign a voltage
vk
and a current ik
in their associated
reference directions. That is,
we always define a branch current
in the direction from the positive to
the negative terminal of the branch.
We then assign, arbitrarily, four voltage
values and four current values
to these four branches, such as the
values shown in the top two rows of
Table 1. We note that the assigned
voltages and currents have nothing
to do with the voltages and currents
of the original circuit, nor are they
TABLE 1. TWO SETS (ONE IN THE BLUE CELLS AND ONE IN THE GREEN CELLS)
OF VOLTAGE AND CURRENTS FOR THE BRANCHES OF THE GRAPH SHOWN
IN FIGURE 1(B).
K = 1
vk (V)
ik (mA)
vˆk
iˆk
(V)
(mA)
4
−5
9
8
K = 2
3
5
1
−8
K = 3
1
1
8
−2
K = 4
1
4
8
−6
IEEE SOLID-STATE CIRCUITS MAGAZINE
FALL 2023
11
|vi
k
kk
+
+
-
-
https://orcid.org/0000-0003-0970-6897
IEEE Solid-States Circuits Magazine - Fall 2023

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