428 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 4, APRIL 1998

Transmission-Line-Based GyratorDoron Shmilovitz, I. Yaron, and Sigmond Singer, Member, IEEE

AbstractA new approach to efficient gyrator realization isdescribed. The approach was motivated by the observation thata =4 segment of transmission line exhibits gyrator-like behavior.A gyrator that operates at low-frequency band, including dc, isformed by the combination of a =4 segment of transmissionline and a switching network. Using a switching bridge, the low-frequency signal is converted into a high-frequency wave whichis applied to the input terminals of the transmission line. Thenthe original waveform is reconstructed using a second switchingbridge. Since the switching bridge, as well as the transmissionline, are loss free (in principle), the resulting gyrator is alsolossless. Such a gyrator can find applications in power processingcircuits. A circuit based on this approach was constructed andtested at the energy conversion laboratory, Tel-Aviv University,Israel.

Index Terms Distributed line topology, gyrator, high-fre-quency converter.

I. INTRODUCTION

THE gyrator is a passive lossless, storageless two-portnetwork defined by the matrix(1)

(see Fig. 1). A main property of the gyrator is that it convertsa one-port network into , its dual with respect to thegyration conductance , defined by [1]:

Various methods of gyrator realization have been suggested,some of them very interesting, such as the Hall-effect gyrator;but the most practical are those based on transistors or someother active devices arranged to operate as linear amplifiers.It has been found by many researchers that using two linearamplifiers and a few resistors, one can synthesize an idealgyrator. Small-signal modeling of gyrator, realized by suchkinds of circuits, is discussed in [1]. The poor efficiency oflinear amplifiers (less than 0.5) results in a poor efficiencyof a gyrator realized by that method. The power losseshave no significant importance as long as the gyrator isapplied for signal processing (which was and still is its usualManuscript received October 2, 1997; revised February 4, 1997 and July

1, 1997. This work was supported in part by the Israel Science Foundation.This paper was recommended by Associate Editor A. Ioinovici.D. Shmilovitz was with the Faculty of Engineering, Tel-Aviv University,

Ramat Aviv, 69978 Tel-Aviv, Israel. He is now with the Department ofElectrical Engineering, New York Polytechnic University, Brooklyn, NY11201 USA (e-mail: shmilo@zpower.poly.edu).I. Yaron and S. Singer are with the Faculty of Engineering, Tel-Aviv

University, Ramat Aviv, 69978 Tel-Aviv, Israel.Publisher Item Identifier S 1057-7122(98)02106-0.

Fig. 1. Gyrator model.

Fig. 2. A transmission line segment coupling two networks.

application); however, they cannot be ignored in the case ofpower processing applications.It has been found that efficient circuits, which have gyrator

characteristics, may be applied in the power processing area.Some of the possible applications are coupling of dc voltagesources with reduced sensitivity of the operation point to vari-ations, ac/dc conversion, and capacitive VAR generation [2].Realization of loss-free gyrator (in principle) can be achievedby the continuous control of a time variable transformer [2].It has also been found that switched mode converters whichnaturally exhibit gyrator behavior do exist [3][5].In this paper, a new type of gyrator, based on the natural

properties of transmission lines, is described. The well-knownfact that a transmission line transfers a short circuit into anopen circuit and vice versa, and actually transfers linear loadslike a gyrator, motivated this approach. Other transmissionline-based power processing networks, like transformers andloss-free resistor, have been reported lately [6], [7].

II. THE GYRATOR-LIKE CHARACTERISTICOF A TL IN THE CASE OF HARMONIC

EXCITATION (HIGH-FREQUENCY EXCITATION)Let us look at a segment of a uniform, loss-free transmission

line (TL) (Fig. 2). It is well known that for such TL, thecharacteristic impedance and the propagation velocity aregiven by

(2)

The propagation time , along a segment of TL whose lengthis , is

(3)10577122/98$10.00 1998 IEEE

SHMILOVITZ et al.: TRANSMISSION-LINE-BASED GYRATOR 429

Fig. 3. =4 TL segment in cascade with a phase shifting network imple-ments a gyrator.

In the case of harmonic excitation at an angular frequency ,the input/output phasors relationship is given by

(4)

where and are the input voltage and current phasors,while and are the output ones. In the case in which thelength of the TL segment is (4) becomes

(5)

A passive impedance, , connected to the output of theTL segment would be seen by the source at the input as itsreciprocal multiplied by which is a gyrator property.

(6)

On the other hand, (5) is the transfer matrix of a gyrator withgyration conductance connected in cascade with a phaseshifting network, so the chain matrix of the TLis given by

(7)(8)

For example, a voltage source is transformed into a currentsource with a phase shift of

(9)

That implies that only reactive power (but not real power)can be transferred between two, in phase voltage (or current)sources. Thus a pure gyrator can be achieved by the cascadeconnection of a phase shifting network , that would cancelthe phase shift, and transmission line segment, see Fig. 3.

III. SQUARE WAVE EXCITATIONIn this section, it will be shown that a transmission line

segment exhibits gyrator-like behavior in the case of squarewave excitation as well.Let be a voltage source with a square waveform which

is applied to the input terminals of the TL; is defined asfollows:

(10)

(11)

Fig. 4. The network equivalent for each of the harmonics.

The definition of implies

(12)

The Fourier series expansion of is

(13)

(14)

assuming the length of the TL is where is thewavelength of the square wave. Substitution into (4) leadsto the following relationship concerning the th harmonicinput/output current/voltage characteristics:

(15)Since

(16)

Substituting

(17)(15) can be written as

(18)

Equation (18) is the equation of a cascade connection of agyrator and a phase shifting element. Thus, by adding to thetermination of the TL, an element whose chain matrix is givenby (19), the system (Fig. 4) exhibits gyrator behavior for eachof the harmonics.

(19)

Thus, such a system can be viewed as a gyrator for a squarewave excitation.

IV. A DC GYRATORIn this section, it will be shown that the chain matrix of

the total system, , is a gyrator matrix when operated withdc sources as well. In particular, it will be shown that it ispossible to couple two dc voltage sources using a transmissionline-based gyrator.A dc voltage can be converted into a square wave by

applying a switching bridge as shown in Fig. 5. In this case, the

430 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 4, APRIL 1998

Fig. 5. The switching cell.

(a)

(b)Fig. 6. (a) The gyrator network and (b) its equivalent.

input/output voltage and current parameters at the switchingbridge terminals are given by

(20)

where

.

(21)

A gyrator suitable for coupling dc sources and loads can beobtained by the cascade connection of a switching bridge,a transmission line, and an additional switching bridgewhose switching function is shifted by in time, as shownin Fig. 6.The relationship between the input/output parameters and

those which appear on the terminals of the transmission lineis given by

(22)

(23)

where are the voltages at the input and output termi-nals of the TL, while are the voltages at the input andoutput terminals of the total system, respectively.Let us assume that each of two, constant voltage sources,and , is coupled by switching bridges

to the input and output terminals of a transmission linesegment, respectively, as in the network shown in Fig. 6.

Using the Fourier series expansion, we obtain

(24)

Substituting , is given by

(25)

since , and where , wehave

Using (4), the following relationship is achieved for the thharmonic:

(26)Substitution of the Fourier expansion (24) for andyields

(27)

(28)

where

(29)Equations (27) and (28) can be written as

(30)

(31)

Using (13) we have

(32)

(33)

But

(34)(35)

(36)

SHMILOVITZ et al.: TRANSMISSION-LINE-BASED GYRATOR 431

(a)

(b)Fig. 7. (a) The equivalent circuit and (b) the simulation layout.

Fig. 8. Simulation results, steady state.

The definition of implies

(37)thus,

(38)

Since the chain matrix of the total system, , is a gyrator ma-trix, it is concluded that the system exhibits gyrator behaviorin the case of dc excitation, i.e.,

(39)

V. GYRATOR OPERATION WITH AC SOURCE EXCITATIONOne of the interesting applications of gyrators is with ac

sources; let us consider the case where the input and output

Fig. 9. Simulation results, transient.

Fig. 10. The experimental setup.

voltage sources are of the form

(40)where is the ac source frequency. Applying the switchingnetworks, the input and output voltages of the TL are created:

(41)

Reformulating (41):

(42)

432 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 4, APRIL 1998

(a) (b)

(c) (d)Fig. 11. Experimental results.

Fig. 12. The equivalent circuit in terms of parasitic resistors.

Incorporating the approximation at the begining of this section,it can be proved that the network has gyrator characteristic foreach of these terms, provided that

(43)

In conclusion, the network is a gyrator with respect to theinput and output ac sources.

VI. THE INFLUENCE OF SLIGHT MISMATCHINGAND GYRATOR OPERATION WITH AC SOURCES

Through the derivations so far, total matching was assumed,i.e., . A sensitivity analysis is required to find out whathappens if a slight mismatching, originated either by the TLlength or by the switching frequency, exists. Does the modelcollapse?Let us consider the case of harmonic excitation where the

angular frequency is slightly different, i.e.,

(44)

Assuming the TL length is tuned to when the excitingfrequency is , in that case,

(45)and the following approximations are valid:

(46)in which case (4) transforms into (47) rather than (5)

(47)If this kind of network is to possess gyrator characteristics, thefollowing condition needs to be fulfilled:

(48)

In conclusion, the network is quite tolerant to length variations;or, viewed the other way, to variations in the switchingfrequency.

SHMILOVITZ et al.: TRANSMISSION-LINE-BASED GYRATOR 433

VII. SIMULATION RESULTSSimulations have been performed for the topology in Fig. 7,

where small filters have been added at the input andoutput of the gyrator. The switching frequency is 100 kHz.Since the load is a capacitor, the current is expected torise linearly.In Fig. 8, the input and output currents and the capacitor

voltage are presented for the steady state, while in Fig. 9, wehave focused on the transient behavior. It can be seen thatthe currents reach steady state within 300 s, which indicatesthat this gyrator can easily operate with line voltages whosefrequencies are 50, 60, or even 400 Hz.

VIII. EXPERIMENTAL RESULTSDue to some drawbacks of TL (volume and ohmic losses),

the experimental setup was based on a -approximation ofTL, like the one shown in Fig. 10. Capacitors and wereadded to eliminate source inductance effects at the switchingfrequency. Since in this switching method, and containdifferent dc components, a capacitor had to be insertedto eliminate the dc coupling; due to its high value it has noinfluence at the switching frequency.The switches were implemented by power Mosfets MTM

15N50 H (toroid core MPP B-166 151-2),F, F, F, the phase shift

was , the switching frequency 50 kHz, V, andthe load resistor was varied to achieve output voltage in therange of 1218 V. For a short circuit at the output, the inputcurrent was 0.1 A, while for output voltage of 15 V the inputcurrent was 0.85 A.The total efficiency was over 77%, the maximum power

was 40 W. and were chosen relatively high in orderto examine the dc response of the static (dc) experiment. Theexperimental results are presented in Fig. 11. Combining theresults of all those graphs, the equivalent circuit, shown inFig. 12, can be derived. The equivalent resistors values were

and .

IX. DISCUSSIONA new TL-based approach for gyrator implementation was

described. This kind of gyrator is capable of operation withlarge signals, thus expected to apply for power processingapplications. The TL-based gyrator can operate with ac signals,as well as dc signals and, in principle, has no losses. Theapproach was verified by simulation as well as by experiments,the results are in good agreement with the expectations.A phase shift, between the two switching networks in

Fig. 6 might deviate from it looks as if a variationof might control the power flow. However, additionalresearch is required to achieve a complete understanding ofthis mechanism.A second unsolved question is the T-LC link approximation

of the transmission line (Fig. 10). For practical applications, itis needed to define the number of links that is required in orderto achieve a good approximation. The values of the capacitorsand inductors in each link have to be defined as well.

REFERENCES

[1] A. N. Willson and H. J. Orchard, Realization of ideal gyrators, IEEETrans. Circuits Syst., vol. CAS-21, pp. 729732, Nov. 1974.

[2] S. Singer, Loss free gyrator realization, IEEE Trans. Circuits Syst.,vol. CAS-35, pp. 2634, Jan. 1988.

[3] M. Ehsani and I. Husain, Power converters as natural gyrators, IEEETrans. Circuits Syst., vol. 40, pp. 946949, Dec. 1993.

[4] S. Singer and R. W. Erikson, Canonical modeling of power processingcircuits based on POPI concept, IEEE Trans. Power Electron., vol.PE-7, pp. 3743, Jan. 1992.

[5] M. Ehsani and I. Husain, Inverse dual converter (IDC) for high-power DCDC applications, IEEE Trans. Power Electron., vol. 8, pp.216222, Apr. 1993.

[6] S. Singer and D. Shmilovitz, The transformer matrix of some trans-mission line topologies, in IEE Proc. Circuits, Devices Syst. Feb. 1995,vol. 142, no. 1, pp. 2126.

[7] , Transmission line based loss-free resistor, IEEE Trans. Cir-cuits Syst., vol. 41, pp. 120126, Feb. 1994.

[8] R. W. Newcomb, The semistate description of non linear time-variablecircuits, IEEE Trans. Circuits Syst., vol. CAS-28, pp. 6271, Feb. 1981.

[9] A. Antoniou and K. S. Naidu, Modeling of a gyrator circuit, IEEETrans. Circuit Theory, vol. CT-20, pp. 533540, Sept. 1973.

[10] P. T. Couse, Broadband transformer design for RF transistor poweramplifiers, in Proc. Electron. Components Conf., 1968, pp. 207216,.

[11] J. Sevick, Simple broadband matching networks, Qst., vol. 60, pp.2023, 1976.

[12] W. J. Sarjeant and R. E. Dollinger, High Power Electronics. New York:TAB Books, 1989, pp. 117136.

[13] M. Macdonald, Design broadband passive components with ferrite,Microwaves RF, pp. 8187, Oct. 1993.

Doron Shmilovitz received the B.Sc. and M.Sc.degrees from Tel-Aviv University, Israel, in 1986and 1993, respectively.In 19961997, he was a Lecturer at Holon In-

stitute of Technological Education. From 1986 to1990, he was an Engineer at I.A.F. Labs. Cur-rently, he is working on postdoctoral studies atNew York Polytechnic University, Brroklyn. Hisresearch interests include modeling and simulationof power converters, and high quality converters andrectifiers.

I. Yaron received the B.Sc. and M.Sc. degrees fromTel-Aviv University, Tel-Aviv, Israel, in 1985 and1995, respectively.Since 1991, he has been working with I.A.I. Labs

as a Software Engineer. He is active in the area ofcircuits and systems and software.

Sigmond Singer (M92) received the B.Sc. andD.Sc. degrees from the Technion, Haifa, Israel, in1967 and 1973, respectively.In 1978, he joined the staff of the Faculty of

Engineering at Tel-Aviv University, Israel, wherehe is currently a Professor. His research interestsare general circuits and systems theory, power elec-tronics, and energy conversion.Dr. Singers paper Realizaiton of Loss Free

Resistive Elements was awarded the 1990 Darling-ton Award by the IEEE-CAS Societys Best PaperAward Committee.