Little known GY - Suppose that you were asked to name the...

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Unformatted text preview: Suppose that you were asked to name the “passive” elements that you can-encounter in studying net- work theory. By a passive element, of course, we mean one with the property that the total energy sup- plied to it has never gone negative. Your answer would probably be re- sistors, capacitors, inductors and transformers, for all of these ele- ments have the property of passivity. There is, however, another passive element, about which we hear much less frequently: the gyrator. It has some surprising, interesting and use- ful properties, and these will be the subject of this article. Let us start out our study of the gyrator by trying to get a clear idea of just what—thiscircuit element is. To developtouragidea, let us first con- sider the'iSubject of analogies. Sup- pose we make an analogy between voltage U in electrical networks and velocity u in mechanical (transla- tional) systems. If we also make an analogy between current i in electri- cal networks and force f in mechani- cal systems, then we can compare corresponding mathematical rela- tions involving these quantities in ' the electrical and mechanical sys- Lawrence P. Huelsman received his BSEE from Case Institute of Technology in 1950 and his MSEE and PhD from the University of California (Berkeley) in 1956 and 1.960 respec- tively. He was employed by Western Electric and taught at the University of California be- fore going to the University of Arizona, where he is currently-aiProfessor of EE. He is an author of a number of technical papers and a recent book on circuit theory, and he is a Senior Member of the. IEEE. 30 terns. For example, a capacitor is characterized by the relation -_ d_v 1_Cdt (1) The analogous equation for a me- chanical system is du f _ M dt (2) This latter may also be written in terms of the acceleration a as f = Ma. This is just one of Newton’s Laws, with the constant M repre— senting the mass of the mechanical element. Thus, comparing (1) and (2) we may say that for the analogies between the electrical and mechani- cal varieties v, u, i, f, that we have postulated, capacitance and mass are analogous physical quantities. Of course, if we had postulated some different set of analogous variables, then we would have gotten a differ- ent set of analogous physical quan- tities. These electrical to mechanical analogies may be easily extended to other network elements. For ex- ample, consider an inductor. It is defined by the relation* i=%fvdt '(3) The analogous equation for a me- chanical system is f=K fu dt (4) * For simplicity of representation we have not shown the limits of integration and the presence of initial conditions, although these must, of course, be considered in an actual computation. The Little The above equation may also be written in terms of distance x as f = Kx. Thus we may say that for the analogies between the electrical and mechanical variables that we have established, inductors (in a recipro- cal sense) and springs are analogous physical quantities. A mechanical analogy for a resistor is also easily found. From Ohm’s Law we have i=Gv (5) Similarly, in mechanical systems we have f=Du (6) where D is a dissipation constant related to sliding friction. Thus electrical conductance and mechani- cal friction are analogous physical quantities. The above analogies and the corresponding symbols for the electrical and mechanical elements are tabulated in Fig. 1*. The/electrical elements that we have found analogies for in the above paragraph are all two-terminal ele- ments. If we wanted to carry our system of analogies even further we might investigate to see if analogies exist for electrical elements with larger numbers of terminals, and mechanical elements with more de—. grees of freedom. We have no assur- * For a more complete discussion of analogies, see D. C. Thorn, An Introduction to Gen- eralized Circuits, Chapt. 1, Wadsworth Pub- lishing 00., Belmont, California, 1963. IEEE STUDENT JOURNAL l l l l i E E i i, ance that such analogies exist, but we may be able to use our ingenuity to discover some of them. First, let us start 'with a known electrical de- vice, the transformer, and see if we can find a mechanical analogy for it. The ideal transformer may be con- sidered as a two-port device, i.e., one with two sets of terminal pairs, defined by the relations v1=nv2 7 i1:(1/n) i2 ( ) Where the variables 11,, v2, and i2 are shown in Fig. 2. The analogous equations for a mechanical system are eul=ku2 f1 = f2 (8) This set of equations is easily recog- nized as defining a simple lever and fulcrum system as shown in Fig. 3. Thus we may say that the simple lever is an analogy of the ideal transformer. Now let us try to apply our anal- ogy seeking process starting with a mechanical element. As an example, let us try to find the electrical anal- ogy of an idealized mechanical gyro- scope. Such a device and an appro- priate set of variables is shown in Fig. 4. For the indicated direction of rotation, the relations among the variables are f,=-ku2 f2 : -ku, (9) JANUARY 1966 Known GletO?” The analogous set of relations de- scribing an electrical network ele- ment is i = g I 1 2 (10) 12 = 'g V1 We shall call the two port network element described by the equations (10) a gyrator1 and give it the circuit symbol shown in Fig. 5. The constant g will be referred to as the gyration conductance. What sort of a device is the gyra- tor and what are its properties? Up l<+¢__, / By L. P. Huelsman to this point we can only say that it is the electrical analogy of a me- chanical gyroscope. Now let’s take a more detailed look at some of its electrical properties. The first prop- erty of a gyrator that we might ex- plore is whether it is a passive or an active device. The instantaneous power consumed by any two-port device is given by the expressiOn P = vli1 + vzi2 (11) If we substitute the relations of (10) into the above expression for power we find that the power supplied to f=M—” f=Kfudt I f=Du Fig. l—Analogies between electrical and mechanical elements. 3] il , I —> nel <— C l 2 + + Fig. 2—The ideal transformer. 1:2 Tb—k—fi+—l l fI A Fig. 3'—A simple lever and fulcrum. 1‘2 Fig. 4—A mechanical gyroscope. the gyrator isidentically zero at all instants of time. Thus, the energy supplied to it is always zero, and we see that the gyrator is a passive net- work element.* As such, it is a new addition to our collection of passive network elements. This is interest- ing, but not of enough significance to make us want to rush right out and start building gyrators. Let’s look at some other properties. Since the gyrator is a two-port network element, we might try ap- plying some two-port network theory to it to determine some of its other properties. For example, consider the y parameters of a general two- * We shall find that the circuit that realizes a gyrator invariably requires active elements, and thus consumes power. The passivity referred to here is with respect to the power supplied to the signal being processed by the gyrator. 32 port network. These are defined by the equations 11(8) = yn(s) V,(s) + y12(s) V2(s)12 12(8) = y-Ms) V.(s) + y22(s) V2(s) where 1,, 1.3, V1, V2, and the yij all are functions of the complex frequency variable 3. By transforming the de— fining equations for a gyrator given in (10) we see that the y parameters of a gyrator are simply yu = 0 y”: g Y21 2"g 372220 Now let’s consider the input admit- tance seen at port 1 of a two-port network when we connect some ter- minating admittance Y2 across port 2 as shown in Fig. 6. In terms of the y parameters of the two-port net- work we obtain (13) YIN = y“ _ LIZ-3% If we now substitute the y parame— ters of the gyrator as given in (13) into the above equation we find that the input admittance is YIN = gg/Yz (15) This is a surprising result! It says that if we connect a capacitor across the terminals of port 2 of our gyra— tor, i.e., if Y2 =’ Cs, then the input admittance is g2/ Cs, in other words, the input admittance is that of an inductor of value C/gg. We now see an application for gyrators, and an important one. Since gyrators will convert capacitors to inductors, we can do everything with resistors, capacitors, and gyrators that we could do with resistors, capacitors, and inductors.* In other words, we can construct inductanceless net- works, using gyrators and capacitors instead of inductors. Furthermore, * Actually, since the gyrator is a two~port device, we can do considerably more; how- ever, a discussion of this would take us be- yond the scope of this paper. For an addi— tional discussion of some gyrator applications see L. P. H uelsman, “Circuits, Matrices, and Linear Vector Spaces,” Ch. 4, McGraw’Hill Book Co., Inc., 1963. since capacitors are usually more ideal than inductors, in the sense that they have less dissipation as- sociated with them, we may be able to produce inductors by gyration with less dissipation, i.e., higher Q, than any actual inductive elements that we could obtain. The weight, size, nonlinearity, etc. of inductors make them unattractive for many network realizations. They are also especially hard to realize in inte- grated circuits, so the gyrator looks as if it should have great promise. One other interesting property of the gyrator can be seen from equa- tion (15). Suppose that we are able to make a gyrator with a very small value of gyration conductance g. For example, suppose that g = 10'*. Then a ten microfarad capacitance will be gyrated into an equivalent inductance of 1000 henries! In other words, there exists the very tantaliz- ing possibility of producing circuit behavior characterized by very large inductors by using very small ca— pacitors. This is especially appealing for compensating networks which must operate at very low frequencies, e.g., in the order of cycles or frac- tions of a cycle per second. From the above properties, it seems as if gyra— tors might have wide application in the field of network theory, if they can be built. So the next question to explore is the possibility of realizing gyrators. ' i i l I g «g d + + VI V2 Fig. 5—The gyrator. —» II” Fig. 6—The driving-point admittance of a terminated network. IEEE STUDENT JOURNAL There are two approaches to the realization of gyrators. One of these approaches uses various types of active network devices. Several such realizations have been presented in the literature. One of these uses an operational amplifier as the active device. An operational amplifier is simply a high gain voltage-controlled voltage source. Thus it has high input impedance and low output impedance. If four operational am- plifiers are connected in the con- figuration shown in Fig. 7, the result— ing two-port network acts as an ideal gyrator2 with a gyration conductance g = 0.2 X 10“. Thus, it can be used to generate circuit characteristics approaching those of very large in- ductors at port 1 when a relatively small capacitor is placed across the terminals of port 2. Another means of realizing a gyrator is through the use of passive components and two NICs (negative-immittance convert- ers). An NIC is a two—port device with the property that the immit— tance seen at one of its ports is the negative of the immittance con- nected to the other port. The prop- erties of such a device were discussed in a previous article in the March 1964 IEEE STUDENT JOURNAL3. The circuit configuration for the gyrator is shown in Fig. 8. One NIC is of the current-inversion type (INIC), the other is used to realize the -R ele- ment.* Other gyrator configurations may also be found in the litera- ture}. 5-6‘ 7 Thus, we see that it is actually possible to realize a gyrator by using existing active devices. Another approach to the realiza- tion of gyrators has also been sug- gested. This is based upon a well- known property of semiconductor materials, the Hall effect. To see * This circuit configuration is usually attrib- uted to H. W. Bode. Actually, the gyrator as a network element was not defined until 1948.‘ In Bode’s book, Network Analysis and Feedback Amplifier Design, D. Van Nostrand Co., Inc., New York, 1945, however, the “ ” configuration of positive- and negative-valued resistors shown at the right side of Fig. 8 appears on page 187. It is the addition of the INI C to this “T” configuration that produces the ideal gyrator. JANUARY 1966 Fig. 7—A gyrator realization that uses operational amplifiers. how this would operate, consider the block of semiconductor material shown in cross-section in Fig. 9. Electrodes are connected to the sides of the block as indicated, so that the resultant terminals constitute a two-port device. The block is placed in a transverse magnetic field into the paper (as indicated by the crosses which represent the tails of arrows). If positive charge flows in the direc- tion of the arrow defining i] (this implies that i1 is positive), then, in passing through the magnetic field, this positive charge will be deflected (the Hall effeCt) to the upper plate, thus producing a positive voltage 122 R R Fig. 8—A gyrator realization that uses negative-immittance converters. at port 2. If, however, positive charge flows in the direction of the arrow defining i2, this positive charge will be deflected so as to produce a negative voltage v1 at port 1. Thus we see that we have the idealized behavior represented by the equa- tions (10) which are the equations defining a gyrator. Actually, due to parasitic effects such as the bulk resistivity of the semiconductor ma- terial, there will also be a non-zero y11 and ygg term in the equations which would have to be eliminated before the device could he considered as an ideal gyrator. This approach appears to have considerable promise for the realization of a gyrator in an integrated form, although com- pletely useful results have not been obtained at this time. Many other uses for gyrators may be found. For example, network transfer functions with the complex conjugate poles which are usually a +_2_ _l:_—_—— in —> XXX XXX + V2 Fig. 9—The Hall-effect gyrator. 33 Fig. l_0—The use of a gyrator to realize a voltage transfer function. associated with circuits that contain resistors, capacitors, and inductors, may besynthesized from resistors, capacitors, and a gyrator. (See Fig. 10.) This circuit realizes the voltage transfer function E __ -s V1 — s2 + 0.2s + 1.01 (16) Another possible application of gyrators is in the production of time-varying elements. For example, a time-varying inductor can be pro- duced by using a fixed capacitor and varying the gyration conductance of the gyrator. Since this gyration con- ductance is usually controlled by 34 resistors, it is relatively easy to vary. This may be considerably simpler than trying to realize a time-varying inductor directly. It is also possible to realize a transformer with a time- Varying turns ratio through the use of time varying gyrators.8 Many other applications remain to be discovered. In the above paragraphs we have introduced a relatively little known and explored network element, the gyrator. Some of the uses to which this element may be put seem to promise an exciting future for it as a relatively important addition to the network designer’s stock of basic elements. It may not only provide a ignor Marconi, right, seems to be contemplating Max- well’s Equations as he poses with his “marvellous ap- paratus” (a spark-gap generator). In the very early years of experimental wireless telegraphy (the Belfast Evening Telegraph quote is a reference to a lecture presented January 20, 1898), this poster appeared in over 600 cities and villages in Great Britain and Ireland. The lecturer, Mr. Lynd, was described as the “Late Principal of West London College of Electrical Engineering.” IEEE Student Branch Program Chairmen should note that Mr. Lynd is no longer available for lectures. Copies of the poster can be obtained from the Marconi Division of The English Electric Corporation, 750 Third Avenue, New York, New York 10017. way for him to accomplish his usual filtering jobs more easily, but it may also provide him with a tool by means of which the relatively un- usual and difficult network synthesis problems that might otherwise prove impossible, can be achieved. REFERENCES ‘B. D. H. Tellegen, The Gyrator, A New Electric Network Element,” Philips Research Reports, vol. 3, pp. 81-101, April 1948. ‘-’ A. S. Morse and L. P. Huelsman, “A Gyra— tor Realization Using Operational Amplifiers,” IEEE TRANSACTIONS ON CIRCUIT THEORY, vol. CT-ll, no. 2, p. 277, June 1964. 3 L. P. Huelsman, “The New Look in Cir- cuit Theory,” IEEE STUDENT JOURNAL, vol. 2, no. 2, pp. 29-35, Mar. 1964. 4 B. P. Bogert, “Some Gyrator and Im- pedance Inverter Circuits,” IRE PROC., vol. 43, no. 7, pp. 793-796, July 1955. G. E. Sharpe, “The Pentode Gyrator,” IRE TRANS. 0N CIR. THEORY, vol. CT-4, Dec. 57, pp. 322-323. 5 T. J. Harrison, “A Gyrator Realization,” IEEE TRANS. ON CIR. THEORY, vol. CT-lO, no. 2, p. 303, June, 1963. 7 M. S. Ghausi and F. D. McCarthy, “A Realization of Transistor Gyrators,” Proc. Nat’l. Elec. Conf., vol. 19, pp. 396—406, 1963. 8 B. D. Anderson, D. A. Spaulding, R. W. Newcomb, “Useful Time—Variable Circuit- Element Equivalences,” Electronics Letters, vol. 1, no. 3, pp. 56-57, May 1965. IEEE STUDENT JOURNAL ...
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