SOME MOTIVATING ARGUMENTS FOR TEACHING ELECTRICAL ENGINEERING STUDENTS

Transcription

1 Far East Journal of Mathematical Education Volume 6, Number 1, 2011, Pages This paper is available online at Pushpa Publishing House SOME MOTIVATING ARGUMENTS FOR TEACHING ELECTRICAL ENGINEERING STUDENTS The Kinneret College on the Sea of Galilee and Braude Academic College, Israel Electrical Engineering Department Ben-Gurion University Beer-Sheva, 84105, Israel 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 Subject 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

2 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 Eastern- European 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.

3 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:

4 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] β v = f () i = Ki, (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 v in( i in ) of the 1-port has a similar analytical structure:

5 SOME MOTIVATING ARGUMENTS FOR TEACHING 69 with β the same as in (1). Here β in = in in in v F( i ) = K i, (2) K in = Kψ( β), (3) with some function ψ () depending on the circuit topology. Because of the feature of νin ~ iin β, 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) ( K ) Kψ ( β), (4) in n = n 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 ν in( i in ) 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 ν in( i in ) 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 m 1 and m 2 placed in a thermostat.

6 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 m 1 have temperature T 1, mass m 2 have temperature T 2, and the temperature of the thermostat be T 0. We assume that T 0 > T1 and T 0 > T 2, which for our analogy corresponds to the fact that the D.C. circuits connected to the voltage source ( v in ) 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 m ( T ) and m ( T ), respectively. 1 0 T1 2 0 T2 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 m ( T T ) + m ( T ). (5) T2 Now, let us, starting from the temperatures T 1 and T 2 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{ T T } < T < max{ T 1, T2}. If the specific heat capacities of the bodies are close, then we can estimate the obtained average temperature T from the equality having ( m + m ) T = m T + m, (6) T2 T = ( m T + m T ) ( m + ). (7) m2 According to the same assumptions that led to (5), the heat flow from the thermostat to the united body is proportional to which, after substituting (7), becomes ( m + m ) ( T ), T ( m + m ) T m T m T = m ( T T ) + m ( T ), (8) T2 1, 2

7 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 fconnected circuits (1-ports) have the conductivity characteristic f ) () and of the ( 1 ( 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 () + f ) (). 1 2

8 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 () = f 1 () + f ( ) + (conductivity characteristics). 2 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, while it is much easier to analyze the composing power-law circuits with the 3 characteristics i = D1v and i = D 3 v. 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. 3

9 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 1 not the conductive, characteristic of the elements, which is the inverse function f. 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.

10 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 q δ( 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 ( v + ) 2. a v b 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.

11 SOME MOTIVATING ARGUMENTS FOR TEACHING 75 Figure 4. Infinite 2D-grid that performs low-frequency filtration of the localized input voltage distribution ( the set { v a, 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 { v v }. See [12]. a, b 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: ϕ ( r2 ) ϕ( r1 ) + ( grad ϕ( r1 )) ( r2 r1 ) (9) that assumes that the vector r 2 r1 is not too large. What is the direction to go from r 1 so that the high spatial harmonics will be enhanced in the potential function ϕ ( r )? e Since the differentiation of kr by r gives the wave-vector (the spatial frequency ) k as a factor, it is obvious that in gradϕ ( r 1 ), the high spatial harmonics are (relatively) enhanced, as compared to those of ϕ ( r ) near r. Thus, 1

12 76 going in the direction (or in the inverse) of the gradient, where the scalar product 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 1 i () t = v( t Δ), (10) R 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 1 i ( t + Δ) = v(). t (11) R

13 SOME MOTIVATING ARGUMENTS FOR TEACHING 77 Using the smallness of Δ, we expand i ( t + Δ), and multiplying by R obtain d Ri () t + RΔ i() t = v(), t (12) dt 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 ω Δ. Since in order to influence the form of the response, ω has to 1 be comparable with Δ, 1 Δ is an important frequency parameter. Thus the system 1 with delay may be used as a filter with a cut-frequency of order Δ, 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 2 see that the derivable from the macroscopic circuit-model expression Li 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 ˆ A t 1 L( e [ ] ) = [ si A], (13) where Lˆ 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 ( ) 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.

14 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 statevariables x : d x = [ 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 d x = F( x, t,...). (15) dt 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.

15 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), [2] E. Gluskin, One-ports composed of power-law resistors, IEEE Trans. on Circuits and Systems II: Express Briefs 51(9) (2004), [3] E. Gluskin, f-connection: a new circuit concept, Proceedings of IEEEI 2008 Conference (Eilat, Israel, 3-5 Dec. 2008), pp (See also my ArXiv works arxiv: and arxiv: ) [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), (See also my ArXiv works.)

16 80 [5] Ch. A. Desoer and E. S. Kuh, Basic Circuit Theory, McGraw-Hill, Tokyo, [6] L. O. Chua, Ch. A. Desoer and E. S. Kuh, Linear and Nonlinear Circuits, McGraw-Hill, New York, [7] E. Gluskin, Spatial filtering through elementary examples, European J. Physics 25(3) (2004), [8] T. Yagi, Interaction between the soma and the axon terminal of retina horizontal cells in cyprinus carpio, J. Physiol. 375 (1986), [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), [10] C. Koch and H. Li, Vision Chips, Los Alamitos, CA: IEEE Computer Society Press, [11] C. Mead, Analog VLSI and Neural Systems, Addison-Wesley, Reading, MA, [12] E. Gluskin, On the ideal low-frequency spatial filtration of the electrical potential at infinity, Phys. Lett. A 338(3-5) (2005), [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), [14] E. Gluskin, Switched systems and the conceptual basis of circuit theory, IEEE Circuits and Systems Magazine, Third Quarter 2009, pp. 56, 58, [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 (See also my ArXive works arxiv: and arxiv: )