h o G Y RAT p COM SIDE RAT I ONS JAY ALLEN WIECHERT B. S., Kansas State University, 1965 A MASTER' S REPORT s u a i 1 1 e d in p a r t i •: 1 f u 1 f i 1 1 men t of t h requirements for the degree MASTER OF SCIENCE Department of Electrical Engineering KANSAS STATE UNIVERSITY Manhattan, Kansas 19 6 6 Approved bv: ^UMMJMJJl Major Professor t.x TABLE OF CONTENTS INTRODUCTION ] 3 PROPERTIES OF GYRATORS AND NETWORKS CONTAINING GYRA 16 SYNTHESIS . THE GYRATOR AS A TOOL IN NETWORK . 25 USING GYRATORS IN MODELING OTHER USEFUL DEVICES A/' GYRATOR REALIZATIONS 50 SUMMARY. ... 51 ACKNOWLEDGMENT 5 2 REFERENCES INTRODUCTION In the Titudy of network analysis and synthesis it is cus- tomary to distinguish between reciproc.il and nonreciprocal net- works. The classification simply depends upon whether or not the network satisfies the Reciprocity Theorem. That is, the ratio of excitation at one port to response at another port is inde- pendent of which of the two ports is excited. If a network contains only resistance, capacitance, self and mutual inductances it is necessarily reciprocal. Networks containing other types of elements are usually nonreciprocal. Quite often the class of nonreciprocal networks is in- correctly identified with the class of active networks. Ord- ir.ariiv active networks are nonreciprocal hut, conversely non- reciprocal networks need not be active. The distinction between active and passive nonreciprocal networks is clarified and studied by adding a fifth basic ele- ment to the list cf conventional passive elements. Although several basic nonreciprocal elements have been proposed the most widely studied is the gyrator, introduced by Tellegen in 1948,' The inclusion of the gyrator permits analysis and synthesis of nonreciprocal networks. Its value has also been demonstrated in the synthesis of reciprocal networks. The ideal gyrator is closely related f. o several other net- 1 work models. The characteristics of these model. ; will be given along with a realization of them using the gyrator. The ideal gyrator has many physical realizations. Several of these are given in the last chapter. Bee a u s e t h e g y r a t o r can be physically realized, it has value, not only as an abstract mathematical model, but also as a practical network element. i PROPERTIES OF CYRATORS Alii) NETWORKS CONTAINING GYRATORS The i d eal gyrator is defined by the symbols in Fig. 1 and by Eqs. 1 and 2 below. V, -R I. (1) V, R I, (2) The constant, real coefficient R has the dimension of ohms and is referred to as the gyration resistance. Notice the linear rela- tionship between variable V and variable I and also between V., and I . The gyrator "gyrates" an output current into an in- put voltage and an output voltage into an input current. Con- versely, the input v a i a b 1 e s are g y rated into the appropriate output variables. If the secondary is short-circuited, the pri- mary terminals appear as an open circuit and vice versa. The gyrator is a passive clement. This may be demonstrated by summing the instantaneous power going into the device at the ports. Non-negative total, ins tan tm nous power into the device indicates a passive network. Let P. and P. be the instantaneous power in at ports one and two, respectively. Then the power into the gyrator at any instant of time is P •!• P . This may be writ- ten in the form P + F = v l + v 3 > l 2 l l 2 *2 < where the small v's and i's represent: instantaneous values of Fig. 1. Gyrator symbol. are sub- and current. If \' and i from Eqs. 1 and 2 voltage 2 2 the stituted into Eq. 3, the total instantaneous power into gyrator is found to be zerc. That is, p + r = v ± + (r^X- vj) x 2 x 1 5 (4) P + P " ° l 2 This indicates that the ideal gyrator is not only passive hut also 1 oss less . The open-citcuit impedance, short-circuit admittance, and coeffi- chair, matrices for the gyrator are given below as the cient matrices in Eqs. 5, 6, and 7 respectively. (5) (6) I-, 'J " v "o R l" "'} (7) 1 1 /'l. E W networks, do The H parameters, quite often used in transistor not exist for the gyrator. As defined the gyrator violates the theorem of reciprocity. The theorem of reciprocity states, that the ratio of excitation the ratio at port one, to response at port two he identical to of excitation at port two to response at port one. More , specifically, if a voltage is applied at port one (V^ = E) and at short-circuited port two (I « I), then the current measured 2 the current measured at short-circuited port one, when the vol- I. This is tage is applied at port two (V = F.) , will also he concisely stated in Eq, 8, (8) V., - With reference to the open-circuit impedance matrix, thii - or h a dual development z » equivalent to stating y.., y,i' 'J ]2 z. .. Equation 6 shows that this condition is not satisfied by } the gyrator. If z „ 4 z for a two-port networl, the network Is said to special case of nonreciprocity occurs when be s (reciprocal. A 2 = -/.,. Networks possessing thin property, of which the gyrator i» nn example, are said to be an tireciprocal . An interesting physical consequence of this property of antireciprocity nay be demonstrated by first loading port two of the gyrator ana examining transmission from one to two (Fig. 2), and by then lo:\ Jim; port one and examining transmission from two from one to two shall be defined to one (Fig. 3) . /Transmission port one only (Vj » E) as V,/V , when a voltage is applied at a applied at porl two only (V ,. I >. Transmission and load 2 from two to one shall likey be defined as V ,/V,, when the same voltage is applied at port two (V, «= El, and the same load ? it be applied at port one (V, - -rl,). ilsinj Eqs . 1 and may verified that transmission from port one to tv.'c, Fig. 2, is as Fi;>. 2 . Gyrator with loaded secondary. Fig. 3. Gyrator with loaded primary. eivon i n Zo (9) 1 Similarly, transmission from two tc one, Fig. 3, is given in Eq. 10. V, (10) The negative sign, present in Eq. 10 and absent in Eq . 9, is an in tic Ltion of antireciprocity. The negative sign in Eq. 10 may he viewed as 180 degrees of phase shift for transmission from port two to pert ope. Equation 9 indicates the absence o£ any ph.- - shift for transmission in the forward direction. Notice that the transmission in the forward and reverse direction differ only in sign and not in magnitude. This re- sults because |z » z and 2 = z != ° £or the l l gyrator. l2 21 l X1 2 ? Transmission in the forward and reverse directions will have the same sign for reciprocal networks. Phase shift in one di- rection will necessarily equal the phase shift in the opposite direction. Although the signs will always be identical the magnitudes, of course , need not be. ^Another very important property of the gyrator is that of impedance inverting. If the gyrator is terminated in an iraDe- Z dance , the impedance looking into pert one will be the re- 2 of Z ciprocal scaled by the factor I; . When a capacitor is 1 impedance be- I rmir.als, the connected to the seconds . Fig. 4, tween the primary terminals is the same as that of an inductor. In Urns of Laplace transform functions, when Z = 1/CS, Z 2 load at port two and Z denotes the R CS where Z denotes the ± ^ 2 input impedance at port one. The impedance R CS is that of an 2 2 poles and nductor of R C henries. In general, Z. = R /?-),- The i n and poles of Z This zeroes of Z are respectively the zeroes ln sane relationship exists when loadin; the primary and observing the impedance at the secondary. Above, the gyrator was considered to be a one-port after information may be loading the other port with Z . Additional two-port gained bj examining the effect of cascading a general two-port net- with a gyrator. It should he expected that the work thus derived would be nonreciprocal due to the antire- ciprocal gyrator. In Fig. 5, the gyrator precedes the general the impedance two-port. The coefficient matrix in Eq . 11 is matrix for the combined two-port network of Fig. 6. Rz a 12 'l h ' 1 '11 0.1) 2 Rz 12 i '11 2 1 J - L J This matrix equation may be verified by noting that the output the general two- voltage of. the gyrator is the input voltage of port and that the output current of the gyrator is the negative indi- of the input current of the general two-port. Equation 11 cates that the network of Fig. 5 is not just nonreciprocal but 10 Fig. A. Gyrator as impedance inverter. ! r -, 4 Z r ll "12 z Z V K. 12 _ 22_ " rig. General two-port and gyrator cascaded, with gyrator on input sid"-. t 12 antircciproc.nl, if the general two-port is assumed reciprocal = network cascaded with one gyrator ( z _ ::..). Any reciprocal will yield an an irecip rocal combination. Notice also, the z , has been inverted and scaled open-circuit input impedance, j 2 by R -just as Z was inverted and scaled in the previous example.