Journal of Circuits, Systems, and Computers
Vol. 19, No. 3 (2010) 529548
#
.c World Scienti¯c Publishing Company
DOI: 10.1142/S021812661000627X
HISTORY AND PROGRESS OF THE TOW THOMAS
BI-QUADRATIC FILTER PART III: GENERATION
¤
USING NAM EXPANSION
AHMED M. SOLIMAN
Electronics and Communication Engineering Department,
Cairo University, Egypt 12613
asoliman@ieee.org
Received 14 April 2009
Accepted 20 October 2009
A new generation method of the Tow-Thomas (TT) circuit based on the nodal admittance
matrix (NAM) expansion is given. The family of TT circuits de¯ned includes four di®erent types
of generated circuits. All the advantages of the classical TT circuit namely, independent control
on the frequency, selectivity factor Q and gain, besides having very low passive sensitivities are
maintained. Besides, each of the family members of the generated circuits use grounded passive
elements and has very high input impedance. It is found that there are eight circuits in each type
implying a total of 32 TT circuits using current conveyors (CCII) and inverting current
conveyors (ICCII). 24 circuits are new and four of them have a °oating property.
Keywords: Admittance matrix expansion; Tow Thomas ¯lter; nullors; pathological mirror
elements; current conveyor; inverting current conveyor.
1. Introduction
One of the most famous active ¯lter circuits is the Tow-Thomas bi-quadratic circuit
(TT).1,2 Part I of this paper3 reviewed the history of TT second-order ¯lter. Two
alternative generation methods of the TT ¯lter were discussed in Ref. 3. Passive and
active compensation methods to improve the circuit performance for high Q designs
were also given in Ref. 3. It was concluded that the classical TT circuit using op amps
has frequency limitations due to the ¯nite gain-bandwidth of the op amps.
Part II of this paper4 reviewed the progress in the realization of the TT circuit
using the operational trans-resistance ampli¯ers, current conveyors (CCII) and the
di®erential voltage current conveyor. The current mode version of the TT circuit
using balanced output current conveyors was also reviewed.
In this paper a new generation method of TT family of circuits based on Nodal
Admittance Matrix (NAM) expansion is introduced.
* This
paper was recommended by Regional Editor Piero Malcovati.
529
530
A. M. Soliman
2. The TT Circuit Using Op Amps
The classical TT active RC circuit using three op amps is shown in Fig. 1.14
Writing Kircho® 's Current Law (KCL) at the inverting input nodes of the ¯rst
two op amps result in:
G4 V1 þ ðsC1 þ G1 ÞV2 G3 V3 ¼ 0 ;
ð1Þ
G2 V2 þ sC2 V3 ¼ 0 :
ð2Þ
From the above equations the transfer functions are obtained as follows:
sG4
C1
V2
¼
;
G2 G3
1
2
V1 s þ sG
C1 þ C1 C2
ð3Þ
G2 G4
V3
C1 C2
¼
:
G2 G3
1
V1 s 2 þ sG
C1 þ C1 C2
ð4Þ
The radian frequency !0 , the selectivity factor Q and the magnitude of the
bandpass and lowpass gains are given by:
sffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G2 G3
1
C1 G2 G3
!0 ¼
; Q¼
;
ð5aÞ
G1
C1 C2
C2
BP Gainð!0 Þ ¼
G4
;
G1
LP Gainð0Þ ¼
G4
:
G3
ð5bÞ
Although the TT circuit using op amps has very low passive sensitivities to all
passive circuit components, it su®ers from a rather drastic Q-factor enhancement
e®ect due to the op-amp ¯nite gain bandwidth.
Several realizations of the TT circuit using CCII5 or ICCII6,7 have been introduced in the literature814 based on block diagram realization. A generalized
G3
G1
G
V1
G4
-
V2
+
2
1
G2
-
V3
+
3
G
+
Fig. 1. The Tow Thomas circuit using three op amps.1,2
-V3
History and Progress of the Tow Thomas Bi-Quadratic Filter
531
generation method based on NAM expansion has not been introduced yet, which is
the main objective of this paper.
3. Derivation of the NAM Equation of TT Circuit
It is desirable to generate a complete family of TT equivalent circuits having exactly
the same equations and using grounded passive elements.
Additionally it is desirable to have very high input impedance for the circuits
belong to the family of voltage mode TT, therefore:
I1 ¼ 0 :
ð6Þ
Using Eq. (6), (1) and (2) can be written as follows:
2 3 2
32 3
0
0
0
V1
I1
4 0 5 ¼ 4 G4 sC1 þ G1 G3 54 V2 5:
0
V3
0
G2
sC2
ð7Þ
The starting point which is the derivation of the TT NAM equation is concluded
by having Y matrix given by:
2
3
0
0
0
Y ¼ 4 G4 sC1 þ G1 G3 5:
ð8Þ
0
G2
sC2
The above Y matrix is de¯ned as type-A, admittance matrix as given in Table 1.
It is worth noting that due to the negative term in the 2, 3 position, the nullor
elements cannot be used alone to get a physical realization of the TT circuit using
grounded passive elements. This demonstrates the importance of the pathological
mirror elements in the systematic synthesis based on NAM expansion.
Table 1. The admittance matrix of the four types of TT circuits.
Type
A
B
C
D
Admittance matrix
"
#
"
0
0
0
G4 sC1 þ G1 G3
0
G2
sC2
0
0
0
G4 sC1 þ G1 G3
0
G2
sC2
"
"
0
0
0
G4 sC1 þ G1 G3
0
G2
sC2
#
#
0
0
0
G4 sC1 þ G1 G3
0
G2
sC2
#
Sign BP
Sign LP
þ
þ
þ
þ
532
A. M. Soliman
Before considering the NAM expansion to obtain the desirable TT family of
circuits, it may be useful to review brie°y the four pathological elements that will be
used.
The nullor elements are the nullators and norators shown in Figs. 2(a) and 2(b).
The nullator and norator are pathological elements that possess ideal characteristics
and are speci¯ed according to the constraints they impose on their terminal voltages
and currents. For the nullator V ¼ I ¼ 0, while the norator imposes no constraints
on its voltage and current. Additional pathological elements called mirror elements
shown in Figs. 2(c) and 2(d) have been introduced in Refs. 6 and 7 to describe the
voltage and current reversing actions. The voltage mirror (VM), is a lossless two-port
network element used to represent an ideal voltage reversing action and it is
described by:
V1 ¼ V2 ;
ð9aÞ
I1 ¼ I2 ¼ 0 :
ð9bÞ
The current mirror (CM) is a two-port network element used to represent an ideal
current reversing action and it is described by:
V1
and
V2 are arbitrary ;
ð10aÞ
I1 ¼ I2 ; and they are also arbitrary :
ð10bÞ
Very recently the systematic synthesis method based on NAM expansion using
nullor elements15,16 has been extended to accommodate mirror elements.17,18 This
results in a generalized framework encompassing all pathological elements for ideal
I
+
I
_
V
+
(a)
(b)
I1
I2
V1
_
+
V2
_
+
(c)
_
V
I1
I2
+
+
V1
V2
_
_
(d)
Fig. 2. The pathological elements, (a) Nullator, (b) Norator, (c) Voltage mirror, (d) Current mirror.
History and Progress of the Tow Thomas Bi-Quadratic Filter
533
description of active elements. Accordingly, more alternative realizations are possible
and a wide range of active devices can be used in the synthesis.
3.1. The NAM expansion realization of TT-circuits type-A
The bracket notation will be used here to generate the TT family of circuits. In
this notation, nodes linked by nullators are indicated by brackets linking the corresponding columns in the NAM. Similarly, nodes linked by norators are indicated by
brackets linking the corresponding rows in the NAM.
The bracket notation is also used with pathological VM and CM but using di®erent
types of brackets to distinguish between nullor elements and mirror elements when
both occur in the realization. In the bracket notation of mirror elements, nodes linked
by voltage mirrors are indicated by brackets linking the corresponding columns in the
NAM and nodes linked by current mirrors are indicated by brackets linking the
corresponding rows in the NAM.
Bracket notation is not a mathematical one and thus it is not suitable for analysis.
On the other hand it is more general than the in¯nity notation,1518 because in the
case where there is more than one pair of pathological elements, the bracket notation
allows pathological elements to be paired in di®erent ways resulting in various
equivalent realizations.17,18 In order to have grounded passive element realizations it
is necessary to get the o®-diagonal elements namely G2 , G3 and G4 to be on the
diagonal positions, besides the G3 must appear as G3 .
The ¯rst step is to add a fourth blank row and column to Eq. (8) and then a
nullator is connected between nodes 2 and 4 to move G2 to the position 3, 4 as follow:
ð11Þ
The next step is to connect a norator between nodes 3 and 4 to move G2 to the
diagonal position 4, 4 as follows:
ð12Þ
534
A. M. Soliman
The next step is to add a ¯fth blank row and column and then to connect a
nullator between nodes 3 and 5 to move G3 to the position 2, 5 as follows:
ð13Þ
The next step is to add a sixth blank row and column and then to connect a
norator between nodes 2 and 6 to move G4 to the position 6, 1 and also G3 will be
moved to the position 6, 5 as follows:
ð14Þ
The next step is to connect a CM between nodes 5 and 6 to move G3 to the
diagonal position 5, 5 as G3 as follows:
ð15Þ
History and Progress of the Tow Thomas Bi-Quadratic Filter
535
The ¯nal step is to connect a nullator between nodes 1 and 6 to move G4 to the
diagonal position 6, 6 as follows:
ð16Þ
The above equation is realized in Fig. 3(a) using three nullators, two norators and
one CM and the circuit realization uses two CCII and one CCIIþ as given in
Table 2.
1
2
G1
4
C1
G2
3
6
5
G4
G3
C2
Fig. 3(a). Pathological element realization of TT circuit A-1.
Table 2. The current conveyors used in Type A TT realizations.
Circuit number
Conveyor 1
Conveyor 2
Conveyor 3
Sign BP
Sign LP
Floating
Ref.
A-1
A-2
A-3
A-4
A-5
A-6
A-7
A-8
CCII
CCII
CCII
CCII
ICCIIþ
ICCIIþ
ICCIIþ
ICCIIþ
CCII
ICCIIþ
CCII
ICCIIþ
CCII
ICCIIþ
ICCIIþ
CCII
CCIIþ
CCIIþ
ICCII
ICCII
CCII
ICCIIþ
CCII
ICCIIþ
þ
þ
þ
þ
þ
þ
þ
þ
No
No
Yes
No
No
No
No
No
8
New
New
New
New
14
New
New
536
A. M. Soliman
3.2. The NAM expansion realization of TT-circuit A-2
The ¯rst step is to add a fourth blank row and column to Eq. (8) and then a VM is
connected between nodes 2 and 4 to move G2 to the position 3, 4 as follows:
ð17Þ
The next step is to connect a CM between nodes 3 and 4 to move G2 to the
diagonal position 4, 4 as G2 therefore Y is given by:
ð18Þ
The remaining steps are the same as in the previous case and the ¯nal NAM
equation is obtained as follows:
ð19Þ
The above equation is realizable using two nullators, one VM, one norator and
two CM and the circuit realization uses one CCII, one CCIIþ and one ICCIIþ as
given in Table 2.
History and Progress of the Tow Thomas Bi-Quadratic Filter
537
3.3. The NAM expansion realization of TT-circuit A-3
The ¯rst two steps are the same as in Sec. 3.1. Starting from Eq. (12) and connect a
VM between nodes 3 and 5 to move G3 to be G3 at the position 2, 5, therefore:
ð20Þ
The next step is to add a sixth blank row and column and then to connect a
norator between nodes 2 and 6 to move G4 to the position 6, 1 and also G3 will be
moved to the position 6, 5 as follows:
ð21Þ
Finally a nullator is connected between nodes 1 and 6 to move G4 to the diagonal
position 6, 6 and a norator is connected between nodes 5 and 6 to move G3 to the
diagonal position 5, 5 as follows:
ð22Þ
The above equation is realized in Fig. 3(b) using two nullators, one VM and three
norators and the circuit realization uses two CCII and one ICCII as given in Table 2.
538
A. M. Soliman
1
4
2
G1
C1
3
G2
C2
5
G3
6
G4
IG = 0
Fig. 3(b). Pathological element realization of TT circuit A-3.
An interesting feature of this circuit is that it has a °oating property19 as IG ¼ 0.
There are ¯ve more circuits that belong to type-A, and are not included to limit
the paper length and are summarized in Table 2.
4. The NAM Expansion Realization of TT-Circuits Type-B
Although a noninverting bandpass response cannot be obtained from the TT circuit
using op amps, it is possible however to obtain noninverting bandpass response using
CCII or ICCII, or a combination of both. Modifying Eq. (1) to be:
G4 V1 þ ðsC1 þ G1 ÞV2 G3 V3 ¼ 0:
ð23Þ
Equation (2) is kept unchanged and in this case the lowpass polarity remains to be
opposite to the bandpass polarity and will be inverting.
In this case the Y matrix to be expanded is given by:
2
3
0
0
0
Y ¼ 4 G4 sC1 þ G1 G3 5:
ð24Þ
0
G2
sC2
The TT circuits obtained based on the above NAM equation is de¯ned as type-B
and is included in the second row of Table 1.
4.1. The NAM expansion realization of TT-circuit B-1
The ¯rst step is the same as in the A-1 circuit realization and in this case Eq. (24) is
modi¯ed to:
ð25Þ
History and Progress of the Tow Thomas Bi-Quadratic Filter
539
The next step is to add a ¯fth and sixth blank rows and columns and to connect a
CM between nodes 2 and 6 to move G4 and G3 to the sixth row with positive signs
as follows:
ð26Þ
The next step is to connect a nullator between nodes 3 and 5 and a norator
between nodes 5 and 6 to move G3 to the diagonal position 5, 5 as follows:
ð27Þ
The ¯nal step is to connect a nullator between nodes 1 and 6 to move G4 to the
diagonal position 6, 6 as follows:
ð28Þ
The above equation is realized in Fig. 4 using three nullators, two norators and
one CM and the circuit realization uses two CCII and one CCIIþ as shown in the
circuit number B-1 realization in Table 3.
540
A. M. Soliman
1
4
2
G1
C1
Fig. 4.
G2
3
6
5
G4
G3
C2
Pathological element realization of TT circuit B-1.
Table 3. The current conveyors used in Type B TT realizations.
Circuit number
Conveyor 1
Conveyor 2
Conveyor 3
Sign BP
Sign LP
Floating
Ref.
B-1
B-2
B-3
B-4
B-5
B-6
B-7
B-8
CCIIþ
CCIIþ
CCIIþ
CCIIþ
ICCII
ICCII
ICCII
ICCII
CCII
ICCIIþ
ICCIIþ
CCII
CCII
ICCIIþ
CCII
ICCIIþ
CCII
ICCIIþ
CCII
ICCIIþ
CCIIþ
CCIIþ
ICCII
ICCII
þ
þ
þ
þ
þ
þ
þ
þ
No
No
No
No
No
No
Yes
No
8
New
New
New
New
New
New
14
4.2. The NAM expansion realization of TT-circuit B-7
The ¯rst two steps are the same as in the previous section. Starting from Eq. (25) and
add a ¯fth blank row and column and connect a VM between nodes 3 and 5 to move
G3 to be G3 at the position 2, 5 therefore:
ð29Þ
The next step is to add a sixth blank row and column and then to connect a
norator between nodes 2 and 6 to move G4 to the position 6, 1 and also G3 will be
History and Progress of the Tow Thomas Bi-Quadratic Filter
541
moved to the position 6, 5 as follows:
ð30Þ
Finally a VM is connected between nodes 1 and 6 to move G4 to the diagonal
position 6, 6 and a norator is connected between nodes 5 and 6 to move G3 to the
diagonal position 5, 5 as follows:
ð31Þ
The above equation is realizable using one nullator, two VM and three norators
and the circuit realization uses one CCII and two ICCII as given in Table 3.
There are six more circuits that belong to type-B, and are summarized in Table 3.
5. The NAM Expansion Realization of TT-Circuits Type-C
The type-C realization is achieved by reversing the sign of the V3 term in Eq. (1) and
the V2 term in Eq. (2). Of course this will not a®ect the denominator DðsÞ of the
transfer function as given in Eq. (3). In this case the basic circuit equations are
modi¯ed as follows:
G4 V1 þ ðsC1 þ G1 ÞV2 þ G3 V3 ¼ 0 ;
ð32Þ
G2 V2 þ sC2 V3 ¼ 0 :
ð33Þ
In this case the NAM equation is given by:
2
3
0
0
0
Y ¼ 4 G4 sC1 þ G1 G3 5:
0
G2
sC2
ð34Þ
542
A. M. Soliman
The change of sign in Eq. (33) results in identical signs of lowpass and bandpass
responses and both will be negative as given in third row of Table 1.
5.1. The NAM expansion realization of TT-circuit C-1
Starting from Eq. (34), the ¯rst step is to add a fourth blank row and column and
then a nullator is connected between nodes 2 and 4 to move G2 to the position 3, 4
followed by the addition of a CM between nodes 3 and 4 to move G2 to the diagonal
position 4, 4 as follows:
ð35Þ
The next step is to add a ¯fth blank row and column and then to connect a
nullator between nodes 3 and 5 to move G3 to the position 2, 5 as follows:
ð36Þ
The next step is to connect a norator between nodes 2 and 6 to move G3 to the
position 6, 5 at the same time it will move G4 to the position 6, 1 as follows:
ð37Þ
The next step is to connect a norator between nodes 5 and 6 to move G3 to the
diagonal position 5, 5. The last step is to connect a nullator between nodes 1 and 6 to
History and Progress of the Tow Thomas Bi-Quadratic Filter
543
move G4 to the diagonal position 6, 6 as follows:
ð38Þ
The above equation is realized in Fig. 5 using three nullators, two norators and
one CM and the circuit realization uses two CCII and one CCIIþ as given in
Table 4.
1
4
2
G1
C1
G2
3
6
5
G4
G3
C2
Fig. 5. Pathological element realization of TT circuit C-1.
Table 4.
The current conveyors used in Type C TT realizations.
Circuit number
Conveyor 1
Conveyor 2
Conveyor 3
Sign BP
Sign LP
Floating
Ref.
C-1
C-2
C-3
C-4
C-5
C-6
C-7
C-8
CCII
CCII
CCII
CCII
ICCIIþ
ICCIIþ
ICCIIþ
ICCIIþ
CCIIþ
CCIIþ
ICCII
ICCII
CCIIþ
ICCII
CCIIþ
ICCII
CCII
ICCIIþ
CCII
ICCIIþ
CCIIþ
CCIIþ
ICCII
ICCII
No
No
Yes
No
No
No
No
No
8
New
New
New
New
New
New
14
544
A. M. Soliman
5.2. The NAM expansion realization of TT-circuit C-2
Following successive steps using bracket notation the following equation is obtained.
ð39Þ
The above equation is realizable using two nullators, one VM, one norator and
two CM and the circuit realization uses one CCII, one CCIIþ and one ICCIIþ as
given in Table 4.
5.3. The NAM expansion realization of TT-circuit C-3
The following successive steps using bracket notation the following equation are
obtained.
ð40Þ
The above equation is realizable using two nullators, one VM and three norators
and the circuit realization uses two CCII and one ICCII as given in Table 4. This
is the only °oating circuit in the type-C circuits.
There are ¯ve more circuits that belong to type-C, and are summarized in Table 4.
6. The NAM Expansion Realization of TT-Circuits Type-D
A noninverting bandpass and noninverting lowpass can be achieved by changing the
sign of V1 in Eq. (32), in this case the NAM equation is given by:
2
3
0
0
0
Y ¼ 4 G4 sC1 þ G1 G3 5:
ð41Þ
0
G2
sC2
History and Progress of the Tow Thomas Bi-Quadratic Filter
545
6.1. The NAM expansion realization of TT-circuit D-1
Starting from Eq. (41) and following successive steps using bracket notation the
following equation is obtained.
ð42Þ
The above equation is realized in Fig. 6 using three nullators and three CM.
The circuit realization uses three, CCIIþ as given in Table 5.
1
4
2
G1
G2
C1
Fig. 6.
Table 5.
3
6
5
G4
G3
C2
Pathological element realization of TT circuit D-1.
The current conveyors used in Type D TT realizations.
Circuit number
Conveyor 1
Conveyor 2
Conveyor 3
Sign BP
Sign LP
Floating
Ref.
D-1
D-2
D-3
D-4
D-5
D-6
D-7
D-8
CCIIþ
CCIIþ
CCIIþ
CCIIþ
ICCII
ICCII
ICCII
ICCII
CCIIþ
ICCII
CCIIþ
ICCII
CCIIþ
CCIIþ
ICCII
ICCII
CCIIþ
CCIIþ
ICCII
ICCII
CCII
ICCIIþ
CCII
ICCIIþ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
No
No
No
No
No
No
Yes
No
8
New
New
New
New
New
New
14
546
A. M. Soliman
6.2. The NAM expansion realization of TT-circuit D-7
Starting from Eq. (41) and following successive steps using bracket notation the
following equation is obtained.
ð43Þ
The above equation is realizable using one nullator, two VM and three norators
and the circuit realization uses one CCII and two ICCII as given in Table 5. This
is the only °oating circuit in the type-D circuits.
There are six more circuits that belong to type-D, and are summarized in Table 5.
7. Generalized Conveyor Con¯guration
Figure 7 represents the generalized conveyor realization which includes the 32
circuits of the four di®erent types A to D as special cases as given in Tables 2 to 5.
It is worth noting that the parasitic resistances RX2 and RX3 can be compensated
by subtracting their values from the design values of R2 and R3 , respectively. Also
the parasitic capacitances CZ1 and CZ2 can be compensated by subtracting their
values from the design values of C1 and C2 , respectively.
The only parasitic elements that a®ect the circuit operation are RX1 and CZ3 .
Several Spice simulations have been included in Ref. 20 for the three CCIIþ circuit,
6
X
V2
Z
Conveyor
Y 1
V1
Y
Z
Conveyor
X 2
2
1
4
G1
V3
3
5
C1
G2
C2
Y
Z
Conveyor
X 3
G3
G4
Fig. 7. Generalized conveyor realization of high input impedance TT circuit.
History and Progress of the Tow Thomas Bi-Quadratic Filter
547
demonstrating the circuit practicality up to 100 MHz and the very limited e®ect
caused by RX1 and CZ3 .
It is important to note that the pathological elements in Figs. 3(a) to 6 can be
paired in alternative ways resulting in more circuits. It should be noted, however,
that other pairing may result in circuits having the capacitor C2 connected to the X
port of a CCII or ICCII which will limit the frequency of operation of the circuit due
to its series connection with RX .21
8. Conclusions
A new generation method of the TT circuit based on the nodal admittance matrix
expansion is given. Each of the family members of the generated circuits uses
grounded passive elements and has very high input impedance. All the advantages of
the classical TT circuit are maintained, namely, independent control on the frequency, selectivity factor Q and gain besides having very low passive sensitivities.
It is found that there are 32 TT circuits using CCII and ICCII. 24 circuits are new
and four of them have °oating property. It is pointed out that the generated circuits
are theoretically equivalent and may di®er practically, depending on the conveyor
circuits used. A comparison among the reported circuits is beyond the scope of this
paper which is mainly a step-by-step demonstration of the NAM expansion. To limit
the paper length, the generation equations of only 10 TT circuits out of a total of
32 TT circuits, and only ¯ve pathological circuits are given.
References
1. J. Tow, A step by step active ¯lter design, IEEE Spectrum 6 (1969) 6468.
2. L. Thomas, The biquad: Part I — Some practical design considerations, IEEE Trans.
Circuit Theor. CT-18 (1971) 350357.
3. A. M. Soliman, History and progress of the Tow-Thomas circuit: Part I, generation and
op amp realizations, J. Circuits Syst. Comput. 17 (2008) 3354.
4. A. M. Soliman, History and progress of the Tow-Thomas circuit: Part II-OTRA, CCII
and DVCC realizations, J. Circuits Syst. Comput. 17 (2008) 797826.
5. A. S. Sedra and K. C. Smith, A second generation current conveyor and its applications,
IEEE Trans Circuit Theor. 17 (1970) 132134.
6. I. A. Awad and A. M. Soliman, Inverting second-generation current conveyors: The
missing building blocks, CMOS realizations and applications, Int. J. Electron. 86 (1999)
413432.
7. I. A. Awad and A. M. Soliman, On the voltage mirrors and the current mirrors, Anal.
Integr. Circuits Signal Process. 32 (2002) 7981.
8. A. M. Soliman, New inverting-non-inverting band-pass and low-pass biquad circuit using
current conveyors, Int. J. Electron. 81 (1996) 577583.
9. J. A. Svoboda, Current conveyors, operational ampli¯ers and nullors, Proc. IEEE 136
(1989) 317322.
10. A. M. Soliman, Theorem relating a class of op amp and current conveyor circuits, Int.
J. Electron. 79 (1995) 5361.
548
A. M. Soliman
11. A. M. Soliman, Current conveyor ¯lters: Classi¯cation and review, Microelectron. J. 29
(1998) 133149.
12. A. M. Soliman, Generation of CCII and CFOA ¯lters from passive RLC ¯lters, Int.
J. Electron. 85 (1998) 293312.
13. A. M. Soliman, Current mode universal ¯lter, Electron. Lett. 31 (1995) 14201421.
14. A. M. Soliman, Voltage mode and current mode Tow Thomas bi-quadratic ¯lters using
inverting CCII, Int. J. Circuit Theor. Appl. 35 (2007) 463467.
15. D. G. Haigh, T. J. W. Clarke and P. M. Radmore, Symbolic framework for linear active
circuits based on port equivalence using limit variables, IEEE Trans. Circuits Syst. I 53
(2006) 20112024.
16. D. G. Haigh and P. M. Radmore, Admittance matrix models for the nullor using limit
variables and their application to circuit design, IEEE Trans. Circuits Syst. I 53 (2006)
22142223.
17. R. A. Saad and A. M. Soliman, Use of mirror elements in the active device synthesis by
admittance matrix expansion, IEEE Trans. Circuits Syst. I 55 (2008) 27262735.
18. R. A. Saad and A. M. Soliman, A new approach for using the pathological mirror elements
in the ideal representation of active devices, International Journal of Circuit Theory and
Applications, Online DOI: 10.1002/cta.534.
19. A. M. Soliman, Adjoint network theorem and °oating elements in the NAM, J. Circuits
Syst. Comput. 18 (2009) 597616.
20. A. M. Soliman, Port interchange in voltage mode current conveyor based ¯lters,
J. Circuits Syst. Comput. 7 (1997) 543561.
21. A. Fabre, O. Saaid and H. Barthelemy, On the frequency limitations of circuits based on
second generation current conveyors, Anal. Integr. Circuits Signal Process. 7 (1995)
113129.