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Unformatted text preview: Suppose that you were asked to
name the “passive” elements that
you canencounter 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 currentlyaiProfessor 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 twoterminal 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 twoport 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 twoport
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 twoport
network element, we might try ap
plying some twoport 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) = yMs) 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 twoport
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 twoport net
work we obtain (13) YIN = y“ _ LIZ3% 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 drivingpoint 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 voltagecontrolled
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 twoport 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 (negativeimmittance 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
currentinversion type (INIC), the
other is used to realize the R ele
ment.* Other gyrator conﬁgurations
may also be found in the litera
ture}. 56‘ 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 negativevalued
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 crosssection in Fig. 9.
Electrodes are connected to the sides
of the block as indicated, so that
the resultant terminals constitute a
twoport 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
negativeimmittance 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 nonzero
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 Halleffect 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
timevarying elements. For example,
a timevarying 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 timevarying
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 sparkgap 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. 81101, April 1948. ‘’ A. S. Morse and L. P. Huelsman, “A Gyra—
tor Realization Using Operational Ampliﬁers,”
IEEE TRANSACTIONS ON CIRCUIT THEORY,
vol. CTll, 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. 2935, Mar. 1964. 4 B. P. Bogert, “Some Gyrator and Im
pedance Inverter Circuits,” IRE PROC., vol. 43,
no. 7, pp. 793796, July 1955. G. E. Sharpe, “The Pentode Gyrator,”
IRE TRANS. 0N CIR. THEORY, vol. CT4, Dec.
57, pp. 322323. 5 T. J. Harrison, “A Gyrator Realization,”
IEEE TRANS. ON CIR. THEORY, vol. CTlO,
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. 5657, May 1965. IEEE STUDENT JOURNAL ...
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This note was uploaded on 07/25/2008 for the course ME 457 taught by Professor Rosenberg during the Spring '07 term at Michigan State University.
 Spring '07
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