299
CORRESPONDENCE
The tuning process may be considered as frequency scaling the transfer function T(s). Thus, a tunable transfer function is given by T(ks)
where k, the frequency-scaling parameter, is a function ‘of the controlling quantity. The most simple form of controlling quantity would
be a tuning voltage VT such that k is determined directly by VT.
II. ELECTRONICALLYVARIABLE IMPEDANCEELEMENTS
Fig. 2. A parametriccapacitance.
The first step in the realization of T(ks) is to realize grounded SIS
of the forms Z/k and Z/k2.
IV. CONCLUSIONS
A. Realization of an SI Z/k
Spectral analysis of linear two-port networks consisting of both
time-invariant and periodically time-varying lumped elements was
presented,’ and a general form of the cascade spectral matrix was obtained. Thus the ideas used in the conventional linear time-invariant
two-port network theory can be extended to the case of linear two-port
networks consisting of both time-invariant and periodically time-varying lumped elements.
REFERENCES
(11E. S. Kuh and R. A. Rohrer, TheoryofLinear Act& Networks. San Francisco,
Calif.: Holden-Day. 1967.ch. 10.
121D. G. Tucker, Circuits wifh Periodically VaryingParameters. London, England:
Macdonald, 1964.
[3] V. A. Taft, Electrical Circuits wirh VariableParamerersIncluding Pulsed Control
Systems. Oxford, England: Pergamon,1964.
[4] B. L. Bardakjian, “Computer aided designof periodically time varying linear networks,” M.A.Sc. thesis,University of Toronto, Toronto, Canada, 1970.
I51 C. Kurth, “Spectral matrix for analysisof time-varyingnetworks,” Elec. Commun.,
vol. 39, pp. 277-292, 1964.
[q L. A. Zadeh, “Frequency analysisof variable networks.” Proc. IRE, vol. 38, pp.
291-299,Mar. 1950.
Consider the network in Fig. l(a) where a four-quadrant unity-gain
analog multiplier (AM) is used to obtain a voltage VVT, where v is the
voltage across the simulated driving-point impedance ZT and VT is a
tuning voltage. Then it follows from Fig. l(a) that the current i is given
by (v-vVT)/Z SO that
where we define k = 1 - VT.
B. .Realization of a Square-Scaled Impedance Z/k2
Consider the network in Fig. l(b) where two four-quadrant multipliers, with gains -2 and -l/2, are employed to realize a voltage
( +2vVT-vVT2) so that the current i is given by
i=
V - 2vvT2 + vv$“’ = (1 -
VT)+
Z
Z
: .
giving
Z
Z
ZT
L. T. BRUTON
Abstract-Electronically
ore
the
of
basic
realized
nonlinear
a conventional
means
of
allows
the
function,
The
ence
cutoff
of
employing
network
element.
ladder
scaling
order,
is strictly
oscillators
the
scaling
an
elliptic
low-pass
be
varied
by
and
leads
required
in order
means
ore
to accomplish
branches
or
high-pass
timing
os
simulated
these
of
fun&
multiplier
of
to practical
external
transfer
dependent
realizations
The
circuits;
filter
analogue
frequency
structure
to
analogue
not
All
ladder
quadrant
of
integrated
ate
four
networks.
frequency
arbitrary
available
RC-active
the
RC-active
impedance
method
presently
tunable
by
filters
signals
by
impedances
a tuning
(1
_
vT)2
=
G.
Note that an inverting summer is required in this realization. The
square-scaled impedance of (2) will be referred to as an SSI.
In both of the above realizations it is assumed that the AMs have
both input voltages and the output voltage specified with respect to
ground and that the input port has an infinite input impedance. Currently available microelectronic AMs approximate to this model.
Electronically Tunable Analog Active Filters
tions
=
transfer
voltage.
that
employ
or
refer-
tuning.
I. INTRODUCTION
An interest in electronically tunable RC-active filter structure
stems from the fact that there is a significant number of applications in
which a small number of tunable filters may be used to replace a much
larger number of nontunable (fixed transfer function) filters. Electronic
tuning has the obvious advantage, compared with mechanical tuning,
that remote, automatic, and rapid adjustment of the filter transfer
function is feasible.
The primary purpose of this correspondence is to show that tunable
RC-active filter transfer functions, of arbitrary order N, may be realized by employing scaled impedance (SI) elements in conjunction with
recently proposed RC-active synthesis techniques. The new method
that is proposed has the advantage that it is particularly useful for
realizing a class of low-sensitivity ladder structure transfer functions;
the proposed circuit realizations may be designed using presently available integrated-circuit operational amplifiers and analog multipliers.
Manuscript receivedJune 7. 1971;revisedOctober 24, 1971.
The author is with the Departmentof Electrical Engineering,University of C&ary,
Calgary, Alta., Canada.
III. TUNABLE TRANSFERFUNCTIONS USING
SI AND SSI ELEMENTS
The simulated-inductance approach to the design of RC-active
ladder filter structures has resulted in the realization of high-quality
fixed transfer function filters [ l]-[4]. The passband insertion loss sensitivity of equiterminated versions of the RC-active ladder structures is
low valued [5] and practical work has confirmed the superiority of
simulated-inductance ladder structures for the realization of high-Q
transfer functions [6]. For these reasons, the problem of realizing tunable RC-active /adder structures will be considered.
In general, the transfer function of an RC-active version of an LCR
filter may be written in the form
TLCR(S) = TLCR(SLi, l/s&
Ri)
where L, represents the simulated inductances, Ci the network capacitances, and Ri the network resistances. Thus, a tuned version of a
simulated-LCR active filter exhibits a transfer function given by
TLCR(~S) = TLCR k&,
&
z
1 a)
(3)
where the frequency-scaling parameter obviously implies impedance
scaling of all inductive and capacitive elements Li and G. If we could
find a general class of LCR filters in which all capacitance elements Ci
and all inductance elements Li are grounded, then it would be possible
to realize a tunable filter by means of grounded SI networks of the 1YPO
given in Fig. 1. Unfortunately, the LCR ladder realizations do not
exhibit the property of grounded C and L elements.
A recently proposed class of active networks [4], containing frequency-dependent negative-resistance (FDNR) elements, possesses
300
IEEETRANSACTIONSON CIRCUIT THEORY,MAY 1972
zT-z$jr
zT~~g~“~~+
T
(a)
ZT = -&
(b)
Fig. 1. (a) Realization of scaledimpedance.ZT.
(b) Realization of a square-scaledimpedancezT.
Fig. 3. Completedcircuit realization of a DCR elliptic low-passfilter.
A. Input Voltage Generator Prescaling
i A straightforward SI realization of the so&e hpacitance C. results &.the required SI, given by l/ksC,, b&unfortunately also achieves
an effective scaling of the voltage generator El by the factor l/k. Thus,
tlie direct SI realization of the source capacitance C. results in the
equivalent source circuit of Fig. 2(a) where EI’ = E,{k. That is, the resultant transfer function is given by k-lT(ks) and i~mugnitude as well
as -frequency scaled. To overcome this problem, it is only necessary to
ire&al& E, by the factor k =(l - VT). This may easily be achieved using
the additional prescaling network given in Fig. 2(b).
.- ..A,complete schematic diagram of a normalized tunable DCR lowpass &\liptic transfer function is given ip Fig. 3 where C,, CL, RLI, RLZ,
R& and D are chosen, using the method of [4], to give the required
untuieti cutoff frequency w,,. Since the realization achieves frequency
scaling, the cutoff frequency of Fig. 3 is giten by WOT
where
00
(b)
Fig. 2. (a) Tuned DCR elliptic low-passfilter. (b) Prescatinginput network.
the useful property that all frequency-dependent elements (that is,
capacitors and FDNR elements) may be grounded Thus, the FDNR
network is obtained from the LCR network by scaling every branch
admittance by s; giving a new network, with identical transfer function
to that of the original LCR network, but with the general fo;m
TDCR(S) = Tom
1
1
Ri, szD. 1 z
I
*>
(
where Di is referred to as a D element and the resultant topologically
similar network as a DCR network Thos, a tunable DCR transfer function has the general form
TDCR(~)
=
TDCR Ri, ;(
1
&)
Therefore, a tunable cutoff frequency OOTis completely determined by
the tuning voltage VT and the untuned (Vr=O) cutoff frequency ~0.
The type of transfer function (Butterworth, Chebyshev, elliptic, etc.) is
determined by fixed elements C, and CL, RL~, RL~,‘. ’ ’ , and the D
elements. Thus, passband ripple, stopband attenuation, etc. are independent of tuning voltage. The D element may be realized using generalized impedance converters (GICs) as given in [4].
B. Tuning Range
There is a practical limit to the tuning range that is determitied by
the linear range of the AMs. In general, there will be a maximum magnitude of tuning voltage ~~~~~ beyond which the AMs will have entered
the nonlinear region. Thus, in terms of VT,,,~~and (5) we note that
wo
WoTmax= 1 - VTmax and
t
from which it is noted that tunability is readily obtainable if all FDNR
impedances I/s2Di and all capacitive impedances I/SC< are grounded so
that SSIs and SIs, respectively, may be employed. For example, the
DCR elliptic low-pass realization, proposed elsewhere by Bruton [4],
meets this constraint and consequently may be voltage tuned. Thus, a
tuned version of a DCR low-pass filter is given in Fig. 2(a), where an SI
network is used to achieve the variable grounded terminating capacitive
impedances and an SSI network is used to realize the variable grounded
FDNR D element impedance l/k2s2D.
(5)
OJQT
=-’
WOTmin
=
wo
1+
VTmx
giving a tuning range
Wwhax
-Wcrmin
=
1 +
VTmax
1 -
VTmax
.(6)
Practical circuit v&sions of Fig. 3 have been constructed in the
laboratory using two-amplifier type Al GICs [6] to realize D elements
and microelectronic AMs. The preliminary practical results have resulted in the construction of a voltage-tunable fourth-order elliptic
transfer function with a cutoff frequency that is &ntinuously adjustable
from 270 to 450 Hz by means of a tuning voltage that is varied from
301
CORRESPONDENCE
- 5.0 to +5.0 V. Excellent agreement has been obtained between theory
and practice; initial experimental curves have been published elsewhere [7].
IV. SUMMARY
The use of grounded SI and SSTelements is proposed as a method
by which RC-active ladder filters, of arbitrary order N, may be realized
such that the cutoff frequency W,,Tis tunable. The method does not employ time-varying network elements; the proposed networks are strictly
analog and rely for accuracy on the fact that the ladder realizations
have excellent low-sensitivity properties and that the basic nonlinear
operations, performed by the multipliers, may be achieved to a high
degree of accuracy using present-day analog multipliers.
The concept of employing SI elements to obtain tunability is, of
course, not limited to ladder structures. For example, both capacitance
elements of a capacitively terminated three-terminal gyrator may be
grounded; consequently, the resultant second-order system is electronically tunable by employing SI networks to realize electronically
variable capacitances and hence a tunable biquadratic section. This
method mai be the best technique for realizing tunable bandpass
structures. (Of course, the zero-passband insertion-loss sensitivity
property is not retained.)
The proposed methods of tuning active filters are becoming increasingly significant because the cost of high-performance operational
amplifiers and analog multipliers is rapidly being reduced. Furthermore,
it can be expected that an increasing amount of the proposed circuitry
will eventually be manufacturable on single chips of silicon. The SI
and SSI networks could conceivably be manufactured as single integrated circuits and the GICs are already being manufactured in microelectronic form.
ventional filter characteristics, the output distortion can be reduced to
arbitrarily small values by controlling the passband characteristics. For
instance, in the case of a Butterworth filter the distortion can be reduced
by increasing the cutoff frequency.
Let us assume that a filter has a transfer characteristic H(w, wg)
(e.g., l/[l +(w/w#]). The output of the filter is g(t) and the input is
f(r). The effective cutoff frequency for the filter is the minimum value of
o0 which, when used, keeps
(If@) - g(t)11 = -$IJ,
I f(t) - g(t) I < e
(1)
where E is a specified error value. In this correspondence we shall determine some easily evaluated bounds on ~0. That is, we wish to determine an Wewhich is such that if wO>w,, then IIf(g(t)l(
<e. We then
state that wc is an upper bound on the effective’cutoff frequency of the
filter.
We shall work with linear-phase, or equivalently, phase-corrected
filters. For many, but not all, input signals, these will result in the
smallest effective cutoff frequencies. This is often desirable. The use
of linear-phase filters also allows us to obtain some easily evaluated
results. To avoid having a shift in time we shall assume that the linearphase shift is zero radians.
We shall work with functions of time which satisfy the following
conditions: f(i) is continuous for all time, it has compact support in
the interval [0, T], it has a continuous derivativej(t) everywhere except
at a finite number of points, and the derivative is of bounded variation
wherever it exists. We define a function fit) in the following way :
j(t) = j(t) = $)
where j(t) exists and
REFERENCES
[I] A. Antoniou, “Realization of gyrators using operational amplifiers and their use
j(t) = + [lim j(t) + lim j(t)]
(2%)
t-toin RC-active network synthesis,”Proc. Inst. E/a. Eng. (London), vol. 116, pp.
i-b+
1838-1850,Nov. 1969.
(21H. .I. Orchard and D. F. Sheahan,“Inductorless bandpassfilters.” IEEE J. Solid- when{(t) does not exist at t=lo. We also assume that the total variation
Stais Circuits,vol. K-5, pp. 108-118,June 1970.
.
[3] W. H. Holmes,S. Gruetzmann,and W. E. Heinlein, “Sharp cutoff I&-pass filters of f(t) is bounded by N. That is,
using floating gyrators,” IEEE .I. Solid-Stare Circuits, vol. SC-4, pp. 38-50, Feb.
19h9.
[4] L. T. Bruton, “Network transfer functions using the conceptof frequency-depen-m
dent negativeresistance,”IEEE Trans. Circuit Theory (Corresp.),vol. CT-16, pp.
\
406-408, Aug. 1969.
[5] H. J. Orchard, “Inductorlessfilters,” Electron.Left., vol. 2, pp. 224-225,June 1966.
We shall make use of the following lemma.
[q L. T. Bruton, “Nonideal performance of two-amplifier positive-impedanceconLemma I: Letf(r) have a Fourier transform F(w); then
verters,” IEEE Trans.Circuit Theory,vol. CT-17,pp. 541-549,Nov. 1970.
171-,
“Electronically-tunable analogue active filters,” London 1971 IEEE Jnt.
Symp.ElectricalNetwork Theory, IEEE Dig. Cat. no. 71 C53-CT.pp. 88-89.
The proof of this lemma is given in [2] and will not be repeated here.
BOUNDSON THE EFFECTIVECUTOFF FREQUENCY
OF LINEAR-PHASE FILTERS
Butterworth Filters
Bounds on the Effective Cutoff Frequency
of Linear-Phase Filters
PAUL M. CHIRLIAN
Abstract
of
Ihe
Easily
several
-The
effeclive
AND CHARLES
culoff
frequency
error it introduces. A Chebyshev
applied
classes
bounds
on the
effeclive
of
norm
cutoff
An &h-order linear-phase Butterworth filter has the frequency
characteristic
1
R. GIARDINA
a filk
is used
frequency
is defined
to define
are
Hb,
this
determined
for
In previous papers, the effective bandwidth of a signal was defined
[l]-[3]. We can utilize some of these results to define the effective bandwidth of a filter. Let us consider this. When a signal f(t) is passed
through a filter, its output will, in general, be distorted. For most con-
(5)
2n
1+
error.
of filters.
Manuscript receivedJune 30, 1970: revisedSeptember17, 1971.
P. M. Chirlian is with StevensInstitute of Technology,Hoboken, N. J.
C. R. Giardina is with Fairleigh Dickinson University, Teaneck,N. J.
wo, n) =
in terms
”
0 00
An upper bound on the effective cutoff frequency of the filter is ue. This
is given by
N
&lc = -p-----.
63)
end2 - 2 cos (x/n)
That is, if wO>~,, then (l.f(r)--g(t)\1 <s. Let us demonstrate this result.
Using the Inverse Transform Dirichlet theorem, we have
(If(t)- s(t)11
5 $J,
F(w)
F(w)1+
”
0wo
- dw.
272
(7)