Far East Journal of Mathematical Education
Volume 6, Number 1, 2011, Pages 65-80
This paper is available online at http://pphmj.com/journals/fjme.htm
© 2011 Pushpa Publishing House
SOME MOTIVATING ARGUMENTS FOR TEACHING
ELECTRICAL ENGINEERING STUDENTS
EMANUEL GLUSKIN
The Kinneret College on the Sea of Galilee and Braude
Academic College, Israel
Electrical Engineering Department
Ben-Gurion University
Beer-Sheva, 84105, Israel
e-mail: [email protected]
Abstract
Rejecting the argument of practicality, we argue that one should just
present Electrical Engineering (EE) and Circuit Theory as interesting
subjects and thus attract the proper young students to study EE in the
academy. Some nontrivial motivating examples are suggested, which all
relate also to physics: we consider a thermodynamics analogy for a
specific connection of circuits of the same topology, spatial filtering, the
physical sense of the concept of inductance, a fractal circuit, and an
equational presentation of a typical nonlinearity. The work thus expresses
the opinion that one of the main reasons why the circuit theory is
interesting is its physical foundation, and the material can be used not
only for teaching the EE, also the physics students.
1. Introduction
The non-fashion point of the present communication is that we are not going to
2010 Mathematics Sub ject Classification: 97XX, 97B50, 97C70, 28A80, 93E11, 34A34.
Keywords and phrases: education, fractals, circuit theory, power-law characteristics, spatial
filtering, mathematical aesthetics.
Received September 2, 2010
EMANUEL GLUSKIN
66
explain to the young that Electrical Engineering (EE) is a perspective field in the
sense of job finding and making money. Through the modern sources of
information, including the Internet, the young know all this much better than do the
academic teachers, just as they know everything else about what is practical better
than we do; indeed, they can advise us in this regard.
However, TV and Internet give some information about everything too early,
and non-pedagogically, like delivering some sweets, so that one loses the appetite
for eating a particularly serious lunch, i.e., for serious studies of a certain subject.
Today, in order to enter University, to see the romantic essence of the things, to
devote himself completely to some serious studies and the art of science, one has to
be somewhat naïve and sentimental. However, already from an early age, instead of
having the old-fashional habit of reading good books and acquiring the sensitivity
needed later for becoming a scientifically thinking and spiritually entire person, -the young person just accumulates some practical and fashionable information from
the general information sources, -- the medium in which he lives.
Though we cannot isolate a talented youth or girl from the modern world of
practicality, let us at least give/offer them the same ideals that once attracted
us, demonstrating that going deeply into an academic subject can be interesting
and opens perspectives for the thought. No better way was ever invented; just by
giving interesting examples. Mathematicians and physicists have managed this
perfectly, which is obvious since there are many good popular books about
physics and mathematics (e.g., “Fundamental Physics” by J. Orear, or “What is
Mathematics” by R. Courant and H. Robbins) including many such examples, which
cause one to start to be interested in these disciplines, and which can be advised by
the teacher. Popular books on physics were the motivation for the young Einstein,
who was instructed in the choice of books by a student close to the family.
Unfortunately, this important pedagogical degree of freedom has no parallel in
the Electrical Engineering and System Theory literature. Though some popular
instruction-type books on practical electronics can be found, especially in EasternEuropean literature, there are no good popular books on basic system-science to
recommend the pupils to read. On our opinion, a stress on the physics basics of the
circuit theory and the logic of the development of the theory, should be useful for
such proposed books.
SOME MOTIVATING ARGUMENTS FOR TEACHING …
67
Thus, the best advice would be to encourage either a circuit theorist or a
physicist to write a good popular book devoted to EE and circuit theory basics, but
we cannot go here so far, suggesting a plan for such a book, just offer some
nonstandard topics, partly associated with our research. Of course, these topics
outline only one of the very numerous possible paths to follow, but the point is not
so much what is the path, rather how to go.
2. A Feature of a 1-port that was Never Stressed/Noted Before
Undoubtedly, every new scientific discovery not only sheds light on the future
way of science, but also on what was already done, i.e., the actual way of things is
that we have to start, from time to time, to better understand what was done in the
past, even if these previous results are something to which we are already very well
adjusted.
There are two possibilities to illustrate this position. In the first, the scientific
discovery that causes the change in our understanding, is something really new,
perhaps even revolutionary, and it leads to seeing past results as something wrong or
missing important points. Quantum theory is a good example, and any new
improvement in technology can also be mentioned here.
In the other, more usual, case, the new result is a generalization of a previous
result. Also, in this case, a qualitative possibility or an important or interesting
outlook missed in the past can be found. Let us consider an example of the latter
kind.
Most children know today what fractal is, but not many children know what a
“1-port” is. However, the resistive 1-port is the starting point of circuit theory.
Focusing first only on the circuit structure, and not on the physical nature of the
elements, let us note (this was done for the first time in [1]) that each 1-port is a
basis for the creation of a specific fractal. Indeed, each branch of a 1-port (as well as
of any other circuit) is also a 1-port, and thus we can repeat the structure of the
whole 1-port in each of its branch, and thus continue. Figure 1 gives such an
example for the first recursion step of a circuit whose structure is absolutely clear:
EMANUEL GLUSKIN
68
Figure 1. A fractal circuit (1-port) after the first recursion step. Each 1-port is a basis
for such a specific fractal circuit. We can repeat each time in each branch either the
whole existing structure, or just the initially given structure, or just replace each time
in the given structure f ( ⋅ ) by F ( ⋅ ) (consider Eqs. (1)-(3)). There are also other
possibilities to “play” with the recursion rule and study the changes in the input
resistance, which children may like. See [1, 2] for more details, but ones’ fantasy can
be the best tool here.
It is certainly interesting to observe how the input resistance is changed with the
recursion steps. Of course, these steps (and the very fractal being created) can be
strongly varied. For instance, only some randomly chosen branches can thus be
replaced in each step, and we can thus obtain very different 1-ports. Such a problem,
or, rather, fractal-game, is suitable for computer simulations which the children can
do.
It was noted in [1] that the power-law elements and circuits studied in [2] are
very suitable here, because for the voltage-current characteristic introduced in [2]
v = f (i ) = Ki β ,
(1)
where K and β are some positive constants, relevant to each branch (i.e., all of the
elements are similar), the input (‘in’) characteristic vin (iin ) of the 1-port has a
similar analytical structure:
SOME MOTIVATING ARGUMENTS FOR TEACHING …
β
vin = F (iin ) = K iniin
,
69
(2)
with β the same as in (1). Here
Kin = Kψ(β ) ,
(3)
with some function ψ( ⋅ ) depending on the circuit topology.
β
Because of the feature of νin ~ iin
, preserved with each recursion, such a fractal
circuit is “constructively solvable” [2], i.e., a formula for ψ(β) can be obtained. The
latter is provided by the recursive procedure f → F → f → " . (We use the
same algorithm “→” for determination of the new ‘F’, it to be taken then as a new
‘f in each step). In the context of the recursion, (3) becomes ([1, 2] for details)
( Kin )n = Kψ n (β) ,
(4)
where n is the number of the recursive step.
Though the simple linear case is obviously included here, i.e., β = 1 is
legitimized in (1), the possibility of obtaining these fractals was not noticed until the
reasonability of considering (1) as a generalization of the linear dependence, was
noted in [1]. The possibility of obtaining νin (iin ) for a nonlinear circuit of a
complicated analytical structure (a really very difficult problem) was the reason [1-3]
for focusing on (1), i.e., on the power-law 1-ports with the similar elements. The
flexibility provided by using (1) for finding νin (iin ) forced us to seek effective
applications for the power-law characteristic, and thus such fractals were noticed.
It becomes clear that in the past study of algebraic (resistive and some other)
1-ports, the interesting feature that any 1-port can be a base for a fractal was missed.
3. The f-connection and the Approximate Analytical (Structural) Superposition
Let us continue to examine the power-law characteristic (1), and see that it also
reveals the possibility of an interesting connection between some different circuits of
the same topology.
In order to introduce the idea of the circuit connection, consider first a
simplified thermal system including two masses m1 and m2 placed in a thermostat.
EMANUEL GLUSKIN
70
The situation with the distribution of the voltages in the relevant circuits below is an
analogy to the situation with the temperature in the thermal system.
Let mass m1 have temperature T1, mass m2 have temperature T2 , and the
temperature of the thermostat be T0 . We assume that T0 > T1 and T0 > T2 , which
for our analogy corresponds to the fact that the D.C. circuits connected to the voltage
source (vin ) are passive, i.e., energy will pass from the source to the circuits. Since
this model is an introductory one, let us make some strongly simplifying
assumptions, first of all that the heat flows from the thermostat into the masses are
directly proportional to the differences of the temperatures, i.e., to the products
m1 (T0 − T1 ) and m2 (T0 − T2 ) , respectively.
The proportionality to also the mass means a geometrical constraint of the
model, which can be realized if the bodies are some hollow balls, or some plates of
the same thickness, or are composed of some such pieces. Then, the areas of the
surfaces of the bodies through which the heat enters are indeed proportional to their
masses and thus, the total heat flow inside the two bodies is proportional to
m1 (T0 − T1 ) + m2 (T0 − T2 ).
(5)
Now, let us, starting from the temperatures T1 and T2 of the bodies, “mix”
them (cause a contact), letting them, by themselves, first (before the interaction with
the thermostat) come to a common temperature T, for which, obviously, min{T1, T2 }
< T < max{T1, T2 }. If the specific heat capacities of the bodies are close, then we
can estimate the obtained average temperature T from the equality
(m1 + m2 ) T = m1T1 + m2T2 ,
(6)
T = (m1T1 + m2T2 ) (m1 + m2 ).
(7)
having
According to the same assumptions that led to (5), the heat flow from the
thermostat to the united body is proportional to
(m1 + m2 ) (T0 − T ),
which, after substituting (7), becomes
(m1 + m2 ) T0 − m1T1 − m2T2 = m1(T0 − T1 ) + m2 (T0 − T2 ),
(8)
SOME MOTIVATING ARGUMENTS FOR TEACHING …
71
equal to the total heat flow (5) of the separated bodies. Thus, the union of the bodies
does not influence the total heat flow received by them.
Let us associate our observation related to the thermal system with the
possibility of “mixing” some electrical circuits so that the total input current taken by
them from the battery (the analog of the thermal flux from the thermostat) will be
weakly changed by the “mixing”, or (as in the linear case) unchanged.
The latter problem is considered in [3, 4] where we introduced the “fconnection” of some electrical circuits, -- the “mixing” procedure. This connection
means that we take two circuits of the same topology (and each having the same
elements in all its branches, but the elements are different for the different circuits)
and directly connect all the pairs of the respective nodes. Figures 2 and 3 illustrate
that.
Figure 2. A circuit illustration of the “f-connection”. The elements of one of the “f-
connected” circuits (1-ports) have the conductivity characteristic f (1) ( ⋅ ) and of the
other f (2 ) ( ⋅ ). All the respective nodes are connected in pairs (including “a with a”,
and “b with b”), as is shown for four nodes. The resulting 1-port is of the same
topology, and for its elements f ( ⋅ ) = f (1) ( ⋅ ) + f (2 ) ( ⋅ ).
EMANUEL GLUSKIN
72
Figure 3. A “hardware” illustration to the “f-connection” shown in circuit terms in
Figure 2. The pairs of the respective nodes of the topologically similar circuits are
mutually connected. From two such connections, the input terminals are led out,
creating the port to be connected to outputs a and b of the battery (Figure 2). Being
interested in the input current of the connection, we compare it with the input current
of the usual parallel connection of the two given circuits. (The separated circuits
play the role of the separated “thermal bodies” of the introductory model, while the
f-connection corresponds to the united body.)
While preserving the topology of the initially given circuits, the f-connection
sums (or is additive regarding to) the conductivities of the branches, because the
respective elements appear to be connected in parallel:
f ( ⋅ ) = f1 ( ⋅ ) + f 2 ( ⋅ ) + " (conductivity characteristics).
Passing now from the circuit synthesis to analysis, we note that if a complicated
conductivity f ( ⋅ ) is given for a structure, then we can convert the interpretation,
presenting a given circuit as an f-connection of the two (or more) analytically
simpler (i.e., with a single-power characteristic) circuits of the same topology.
When possible, this interpretation is attractive, because it can be very difficult to
directly analyze a circuit, e.g., with the branch characteristic i = D1v + D3v 3 , while
it is much easier to analyze the composing power-law circuits with the
characteristics i = D1v and i = D3v 3 . The point is that the total input current (our
“thermal flow”) indeed is very little changed by the f-connection of the simpler
circuits, which is because of the following reason that (see also [3, 4]) justifies the
introduction of the concept of the f-connection. The thermal analogy helps one to see
the point.
SOME MOTIVATING ARGUMENTS FOR TEACHING …
73
Similarly to the fact that with the “mixing” of the bodies in the thermal problem
the temperature attains some average intermittent value, the nodal voltages after the
“f-connecting” attain some intermittent values with respect to those that they had in
the separated circuits with the power-law characteristics (we compare, of course, the
respective nodes).
Using this circumstance, let us consider the internal nodes that are close to one
of the input nodes, e.g., to the grounded node b (Figure 2). Obviously, the branch
currents of these few branches collected at the input node compose the input current.
In the f-connection, there are two elements connected in parallel in each of these
branches. Comparing this situation to the initial one where these elements were
separated, each belonging to its “mother” circuit, one sees that because of the
intermittent values of the internal nodal potentials, the currents of these parallel
elements are changed with the f-connection so that one of the currents is increased
and the other is decreased. This causes the total input current to be weakly (see [3,
4] for more details) changed by the connection.
This is the “circuit mechanism” which preserves the input current with an
unexpectedly (for such a strongly nonlinear circuit) high precision, which should be
considered in the theory of resistive (rather, algebraic, since D.C. magnetic and
ferroelectric circuits, or impedance circuits, are relevant too) circuits/networks.
There also is the dual formulation of the f-connection which requires, in
particular, the input current source to be used (i.e., a resistive and not conductive
circuit definition), we should replace nodes by meshes, and the nodal potentials by
mesh currents. Thus, instead of connecting the respective nodes, we now have to
“connect” the respective meshes. That is, now the respective resistive elements
should be connected in series. Thus, the “connection” of all the respective meshes
means their merging into larger meshes including more elements. In the dual
formulation, the “f-connection” remains additive, but with regard to the resistive, and
not the conductive, characteristic of the elements, which is the inverse function f −1.
Last, but not least, since this problem deals with circuits of the same digraph,
one can apply the remarkable Tellegen’s theorem [5, 6] for any needed analytical
treatment of the circuits, which might be useful and interesting to the students.
Computer simulations of such composed circuits are also not very difficult.
Unfortunately, the interesting and easily definable f-connection does not appear
in the classical circuit theory.
EMANUEL GLUSKIN
74
4. A Comment on Spatial Filtering
The concept of spatial filtering has to be introduced into the general teaching
programs. This concept is useful in understanding different physical situations, such
as illumination, or the distribution of electrostatic potential. It is relevant to the idea
of vision chips [7-11], and even to the response of a car to the irregularities of the
road [7].
The possibility of introducing different kinds of symmetries (the central one, or
with respect to a plane, etc.) makes the very physical space around us be a “system”
that can perform spatial filtration. Thus, consider a point charge q at the origin, i.e.,
the spatial charge distribution qδ(r ) , or its potential q r , as the input of a spatial
system, and the resulting (much smoother, obviously) distribution of a potential on a
plane (not including the charge) as the output. In terms of the spatial Fourier
expansion, the main harmonics of the “output” are, obviously, much lower than
those of the input (this simply means that the output function is not so strongly
localized), i.e., we have some low-frequency spatial filtration in the input-output
map.
Work [12] even interprets the constant potential at infinity of a system as the
result of the low-frequency spatial filtration by the system of its voltage input. Thus,
for instance, according to [12], for the system shown in Figure 4 (that can be closed
on a big ball in order to simplify the perception of the “infinity”) the potential at
infinity must be equal to (va + vb ) 2.
Both the low- and the high-frequency filtrations are perfectly demonstrated
using a resistive ladder-circuit with some batteries included in the branches, as the
input. See [7], though this classical example is also found in many introductions to
the modeling the eye-retina action, e.g., [10, 11].
Observe that setting the potential at infinity is also relevant to the quite prosaic
topic of grounding.
SOME MOTIVATING ARGUMENTS FOR TEACHING …
75
Figure 4. Infinite 2D-grid that performs low-frequency filtration of the localized
input voltage distribution ( the set {va , vb }). Take any line of nodes, not including
the input nodes and consider the smoothed distribution of the nodal potentials along
this line as the “output”. The constant potential at infinity is the ideally filtered
average of the input {va , vb }. See [12].
The spatial filtering can also be a high-frequency one. Consider the relation
between electrical potentials (in any electrostatic problem) in two different spatial
points according to the basic formula:
G
G
G G
G
ϕ(r2 ) ≈ ϕ(r1 ) + ( grad ϕ(r1 )) (r2 − r1 )
(9)
G G
that assumes that the vector r2 − r1 is not too large. What is the direction to go from
G
r1 so that the high spatial harmonics will be enhanced in the potential function
G
ϕ( r ) ?
G
G
G
Since the differentiation of e k r by r gives the wave-vector (the spatial
G
G
“frequency”) k as a factor, it is obvious that in grad ϕ(r1 ) , the high spatial
G
G
harmonics are (relatively) enhanced, as compared to those of ϕ(r ) near r1. Thus,
EMANUEL GLUSKIN
76
“going” in the direction (or in the inverse) of the gradient, where the scalar product
G G
G
grad ϕ(r1 ) (r2 − r1 ) is maximal, we obtain an addition of some high-harmonic
components to the potential function, and this is the direction of a high-frequency
spatial filtration. The ladder circuit also demonstrates [7] both the low and high
frequency filtrations.
The academic non-triviality of the topic of spatial filtering and the potential at
infinity is associated, in particular, with the fact that voltage input does not lead us,
in the continuous analogy, to Poisson’s equation for potentials where some currents
are the input(s) (and thus the constant potential at infinity can be chosen arbitrarily),
but to the Helmholtz equation [12, 7] including a non-diffentiated term with voltage
that cannot be “shifted” letting one to arbitrary choose the voltage (potential) at
infinity.
5. Mathematical Phenomenology and Physical Sense … What is an Inductor?
Every electronics engineer knows that a gyrator [6] can turn the capacitor’s
action to that of an inductor. However, it is also useful to know that, for instance,
some not very weak inductive features of a fluorescent lamp [13] operated at the
regular line frequencies, are not of magnetic but of electrostatic origin. This is
associated with some diffusion processes, i.e., with separation of the charges (i.e.,
with electrostatic energy) which just causes the typical inductive feature of a delay of
the current with respect to the voltage [13]. The “true” magnetic energy of the lamp,
associated with a magnetic field is relatively very small.
To see this point better and in a wider scope, it is useful to consider simple
model-equation presenting some current i (t ) delayed with respect to some
associated voltage v(t ):
i (t ) =
1
v (t − Δ ) ,
R
(10)
where R is a resistive-type parameter, and Δ is a small positive time-constant. For
instance, in (10), v (0) → i (Δ ) , i.e., v (t ) appears earlier. Shifting the time origin in
(10), we have
i (t + Δ ) =
1
v (t ).
R
(11)
SOME MOTIVATING ARGUMENTS FOR TEACHING …
77
Using the smallness of Δ, we expand i (t + Δ ) , and multiplying by R obtain
Ri (t ) + RΔ
d
i (t ) = v (t ) ,
dt
(12)
which is identical to the equation for a series R-L circuit with the inductance
L = RΔ connected to the voltage source v (t ). (Because of the delay of i (t ) and
causality, it is physical for v (t ) and not i (t ) to be interpreted as the source; when
expanding v (t − Δ ) in (10), one would have to interpret −i (t ) as the source.)
Such a circuit possesses a frequency-dependent response in which the frequency
ω is included via ωΔ. Since in order to influence the form of the response, ω has to
be comparable with Δ−1, Δ−1 is an important frequency parameter. Thus the system
with delay may be used as a filter with a cut-frequency of order Δ−1, defined by the
delay.
Every physical system (perhaps even a living organism under some applied
voltage, which can be studied in electrical safety science) in which the current is
delayed with respect to the voltage can have some inductive futures, but one has to
see that the derivable from the macroscopic circuit-model expression Li 2 2 can
reflect some electrostatic energy, associated with some charge diffusion and charge
separation.
6. The System-theory Lessons: Speak about the Beauty of the Equations!
After we have developed, while teaching LTI systems, the equality
Lˆ (e[ A] t ) = [sI − A] −1,
(13)
where L̂ is the Laplace transform operator, we should ask the students whether or
not they ever saw such a beautiful equation.
Indeed, we have the number (≈ 2.71828) ‘e’ originating from analysis, matrices
[ A] and [sI − A] −1 from algebra, the complex variable ‘s’ originating from the
great union (the complex plane) of algebra and geometry. It is also remarkable that
[ A] is not a simple algebraic object, but possesses a mathematical structure that
reflects, to a degree, the structure of the real system.
EMANUEL GLUSKIN
78
If we consider with the students the aesthetic side of the equations, then
they will not completely later replace the analytical investigations by numerical
simulations, thus losing the generality and beauty of the analytical outlook, and they
will not forget to check the correctness of the physical dimensions of the formulae
and to analyze the cases of the limit values of the parameters, etc.
7. Nonlinearity and the System’s Structure
Following the above comments on (13), let us somewhat continue with the topic
of the structural presentations, starting from the fact that structural generalizations,
given by the use of the matrices in the linear theory, are one of the main leitmotivs
and attractive points in the linear theory. We widely use linear systems because we
can easily generalize them in the structural sense.
Let us try to keep the structural presentation in a nonlinear case as well. This is
desirable since very different structural generalizations are relevant to modern
electronics technology where miniaturization of a chip is often associated with
repeatability of blocks; this may occur in different VLSI circuits, e.g., such as
“vision chips”. The involved elements/blocks need not be linear, of course.
Following [14], we can try to express the nonlinearity of some systems as a
dependence of the matrices, appearing in the describing equations, on the stateG
variables x :
G
G
G
dx
= [ A( x , t )] x + " .
(14)
dt
While keeping (preserving) the structural matrix form, typical for linear systems, this
nonlinear vectorial equation arises naturally for some switching and sampling
nonlinear systems [14, 15].
Form (14) shows a significant distinction from the classical normal form
G
G G
dx
= F ( x , t , ... ).
dt
(15)
G G
Though (15) is somewhat more general (consider the case of F (0, t , ...) nonzero), it
just suggests going more deeply into the analysis of the given certain system, while
for (14) structural generalizations of the system, required by the engineering
direction of thought, become as natural as for linear systems.
SOME MOTIVATING ARGUMENTS FOR TEACHING …
79
In simple words, equation (14) defines nonlinearity of a system as the influence
of the processes in the system on its structure. Undoubtedly, this is a heuristically
useful outlook on nonlinearity.
Consider, for instance [15], that the velocity field of a liquid flow relates both to
the unknown velocity components to be found and to the “structure” of the liquid
“system”, i.e., we have a situation of (14), i.e., of the kind [ A(x )]. This immediate
observation of the flow is sufficient for concluding that hydrodynamics equations
must be nonlinear, and thus, for instance, for concluding that turbulence is a kind of
chaos obtained in a nonlinear system.
Such equational situations should also be found in sociology, the theory of
differential games and many other fields, and we think that the importance of the
concept of nonlinearity becomes clearer and more feasible when it is possible to pass
on from (15) to (14) and to thus start thinking in structural terms.
8. Conclusions
The opinion is expressed that we should continue to teach in the classical style,
giving the pupils simple and interesting examples with some philosophical
background, letting the pupil to feel that knowledge per se is richness, and that
daring in the field of basic science in not less interesting than any other daring.
Some nontrivial motivating examples are noted, and many, many other
interesting examples can be found by the readers having research and teaching
experiences.
References
[1]
E. Gluskin, On the symmetry features of some electrical circuits, Int. J. Circuit Theory
and Applications 34 (2006), 637-644.
[2]
E. Gluskin, One-ports composed of power-law resistors, IEEE Trans. on Circuits and
Systems II: Express Briefs 51(9) (2004), 464-467.
[3]
E. Gluskin, f-connection: a new circuit concept, Proceedings of IEEEI’2008
Conference (Eilat, Israel, 3-5 Dec. 2008), pp. 58-60. (See also my ArXiv works
arXiv:1004.4128 and arXiv:1004.4428.)
[4]
E. Gluskin, An estimation of the input conductivity characteristic of some resistive
(percolation) structures composed of elements having a two-term polynomial
characteristic, Physica A 381C (2007), 431-443. (See also my ArXiv works.)
80
EMANUEL GLUSKIN
[5]
Ch. A. Desoer and E. S. Kuh, Basic Circuit Theory, McGraw-Hill, Tokyo, 1969.
[6]
L. O. Chua, Ch. A. Desoer and E. S. Kuh, Linear and Nonlinear Circuits,
McGraw-Hill, New York, 1987.
[7]
E. Gluskin, Spatial filtering through elementary examples, European J. Physics 25(3)
(2004), 419-428.
[8]
T. Yagi, Interaction between the soma and the axon terminal of retina horizontal cells
in cyprinus carpio, J. Physiol. 375 (1986), 121-135.
[9]
B. E. Shi and L. O. Chua, Resistive grid image filtering: input/output analysis via the
CNN framework, IEEE Trans. on CAS – Pt. I 39(7) (1992), 531-547.
[10]
C. Koch and H. Li, Vision Chips, Los Alamitos, CA: IEEE Computer Society Press,
1995.
[11]
C. Mead, Analog VLSI and Neural Systems, Addison-Wesley, Reading, MA, 1989.
[12]
E. Gluskin, On the ideal low-frequency spatial filtration of the electrical potential at
infinity, Phys. Lett. A 338(3-5) (2005), 209-216.
[13]
E. Gluskin, The fluorescent lamp circuit, (Circuits and Systems Expositions) IEEE,
Transactions on Circuits and Systems, Part I: Fundamental Theory and Applications
46(5) (1999), 529-544.
[14]
E. Gluskin, Switched systems and the conceptual basis of circuit theory, IEEE Circuits
and Systems Magazine, Third Quarter 2009, pp. 56, 58, 60-62.
[15]
E. Gluskin, A point of view on the linearity and nonlinearity of switched systems,
Proceedings of 2006 IEEE 24th Convention of Electrical and Electronics Engineers in
Israel (15-17 Nov. Eilat), pp. 110-114. (See also my ArXive works arXiv:0807.0966
and arXiv:0801.3652.)