INTRODUCTION
In the literature, it has been presented the design of many active filters,
known as opampRC (Huelsman and Allen, 1980), which have been widely used in
many lowfrequency applications, telecommunication networks, signal processing
and conditioning circuits, communication systems, control and instrumentation.
In some applications, filters with lownoise level and with stable high Q are
strictly desirable to process weak signals. For instance, a lowfrequency opampRC
filter can be required to detect, measure and quantify biomedical signals, object
vibrations, among others. However, as already shown in (Nobuyuki and Nakamura,
2005), to design a lowfrequency filter the use of large capacitances is required
to implement large RC time constants, whose realization with IntegratedCircuit
(IC) technology requires of a large silicon area. This is considered a problem
within the design of analog ICs, e.g., active filters. A solution to this problem
within the design of active filters, is the use of the Operational Transconductance
Amplifier (OTA) (Geiger and SánchezSinencio, 1985), because its implementation
using CMOS IC technology performs low sensitivity, low power consumption, low
noise, low parasitic effects and low cost of fabrication (SánchezSinencio
and SilvaMartínez, 2000).
In this research is introduced a method to transform a VoltageMode (VM) opampRC filter to a GmC filter working in CurrentMode (CM). The case of study begins with the simulation of an already designed lowfrequency opampRC filter with stable high Q (Nobuyuki and Nakamura, 2005), by eliminating one node by applying the transformation YΔ (Chua et al., 1987). The elimination of one node leads us to the formulation of three nodal equations, which are synthesized with OTAC blocks having grounded capacitors, as demonstrated in (Ahmed et al., 2006). This process generates a VM GmC filter, which is transformed to a CM GmC filter by applying adjoint transformations (Torres and Tlelo, 2004). The symbolic expressions of the transfer functions for both, the VM and the CM GmC topologies are derived to demonstrate that both filters perform the same behavior.
OPAMPRC FILTER WITH STABLE HIGH Q
In Fig. 1 is shown a lowfrequency secondorder opampRC filter with stable high Q, which has been already designed in (Nobuyuki and Nakamura, 2005).

Fig. 1: 
Lowfrequency
opampRC filter with stable high Q 

Fig. 2: 
Lowfrequency
opampRC filter with stable high Q, from the transformation YΔ 

Fig. 3: 
Frequency
response of the opampRC filter shown in Fig. 2 
As one sees, the resistances Rya, Ryb and Ryc form a Ytype network, which include an extra node which can be eliminated by applying the transformation Y to Δ (Chua et al., 1987). Indeed, this kind of transformations are quite useful to simplify passive networks, because it reduces the system of nodal equations. The transformed opampRC filter is shown in Fig. 2, where if Rya = Ryb = 19x10^{3}Ω and Ryc = 1x10^{3}Ω, then one gets the network Δ with: Rda = 3.99x10^{5}Ω and Rdb = Rdc = 2.1x10^{4}Ω .
The design of this filter for a central frequency of f_{0} = 795 Hz,
a quality factor of Q = 20 and by using ideal opamps, generates the following
values for the elements (Nobuyuki and Nakamura, 2005): C = C_{2} = 100
pF, R_{1} = 100 KΩ, R_{da} = 3.99x10^{5} Ω,
R_{db} = R_{dc} = 2.1x10^{4} Ω, R_{01}
= 1 KΩ, R_{02} = 99 KΩ, R_{00} = 132 KΩ. The
SPICE simulation is shown in Fig. 3, where V0 is the desired
bandpass response (VoBP) and V01 and V02 are nodes simply referred to VoLP
and VoHP, respectively. The results are in good agreement with Nobuyuki and
Nakamura (2005) using the Δ network instead of the Y one.
TRANSFORMATION OF A VM OPAMPRC FILTER TO A VM GmC FILTER
By beginning with the opampRC filter shown in Fig. 2 and by applying the method shown in (Ahmed et al., 2006), the VM opampRC filter can be transformed to a VM GmC filter by performing two steps: formulation of nodal equations and synthesis of equations with OTAC blocks.
Formulation of nodal equations: The goal is the formulation of the general firstorder equation given by (1) in each node. This equation can be synthesized using OTAC blocks only if (2) is used to replace C_{i} into (1). That way, if C_{i} is equivalent to C_{0}, G_{ci} should be fixed to G_{0}, so that (1) is transformed to (3). The especial cases of (1) can be expressed by (4) and (5).
The formulation of a nodal equation of the type (3)(5) from Fig. 2, begins by the selection of a node which is considered to be the outputvoltage, where the outputs of the opamps have priority, while the remaining nodes are considered to be the inputs to synthesize the nodal equations. In this manner, since in Fig. 2 the node m = V_{02}, because the opamp OP2 is performing a buffer operation, then one can formulate three nodal equations, which are described by (6)(8).
Equation 6 is formulated at the input of OP1, from which
V_{02} can be evaluated. Equation 7a is formulated
at the input of OP2, in this case one should to use (2) according to (7b) to
get the form of (3) which is described by (7c) and from which V_{01}
can be evaluated. Finally, (8) is formulated at the input of OP3, from which
V_{0} can be evaluated. It is important to note that by the properties
of the ideal opamp and from the nodal formulation, Ri and Rdb are cancelled
and Rdc does not affect the system of equations.
Synthesis of nodal equations: In Fig. 4 is shown the synthesis of (6), (7c) and (8) using OTAC blocks. As one sees, the synthesis is performed by accomplishing Kirchhoff’s Current Law (Chua et al., 1987). This process generates circuits with grounded capacitors, which can be implemented into an IC with a smaller area than a floating capacitor, additionally a grounded capacitor can absorb the deviation caused by the shunt parasitic capacitances. The transconductances in each OTAC block are: G_{01} = 1x10^{–3}, G_{00} = 7.5757x10^{–6} and G_{02} = 1.0101x10^{–5}, for Fig. 4a; Ga = G_{01} = G_{c1} = 2.5x10^{–6}, for Fig. 4b and G_{1} = 1x10^{–5}, for Fig. 4c. By joining the synthesized blocks, the VM GmC filter shown in Fig. 5, is obtained.
The symbolic transfer functions of each outputvoltage in the VM GmC filter,
using the method given in (TleloCuautle et al., 2004), are described
by (9a), (9b) and (9c), where V_{0} performs the desired bandpass behavior
and V_{02} and V_{01} are simply associated to VoHP and a VoLP.
The SPICE simulation results of the VM GmC filter are shown in Fig.
6, where the behavior of the VM GmC filter is the same as of the VM opampRC
filter shown in Fig. 3.

Fig. 4: 
(a)
Synthesis of (6), (b) Synthesis of (7c) and (c) Synthesis of (8) 

Fig. 5: 
VM
GmC filter designed from the transformation of the opampRC filter shown
in Fig. 2 
TRANSFORMATION OF THE VM GmC FILTER TO A CM GmC FILTER
In (Koziel and Szezepanski, 2003) one can found several OTAC circuits working
in both VM and CM. On the other hand, Torres and Tlelo (2004) demonstrated the
implementation of adjoint OTAC filters using CMOS IC technology.

Fig. 6: 
Frequency
response of the VM GmC filter, where V_{0} is related to VoBP,
V_{02} to VoHP and V_{01} to VoLP 
For instance, the rules to transform a VM OTAC filter to a CM OTAC filter
and viceversa, are the following (Torres and Tlelo, 2004):
• 
Model the behavior of all OTAs using Voltagecontrolled Current Sources
(VCCS). 
• 
Interchange the ports of each VCCS, but letting intact the positions of
the rest of the circuit elements. 
• 
If a voltage/current source is connected in the input port, this port
is short/open circuited. Now, this becomes to be the output port used to
measure current/voltage. 
• 
If a current/voltage signal is measured at the output port, one should
to connect a voltage/current source. Now, this becomes to be the input port
of the adjoint circuit. 
By executing this steps, one gets the adjoint circuit of the original one.
It is worthy to mention that the symbolic transfer functions of both, the VM
and the CM circuits, should be the same to said that they are adjoints. In this
manner, by beginning with the VM OTAC circuit shown in Fig. 5
and by applying the transformation rules, the adjoint CM OTAC circuit is shown
in Fig. 7. The VM filter has one input and three outputs,
while the CM filter has one output and three inputs. The symbolic transfer functions
are described by (10).

Fig. 7: 
CM
GmC filter designed from the adjoint transformation of Fig.
5 
For the CM OTAC filter, it is possible to add the three responses described
by (10) to derive (11). Besides, the derived symbolic transfer functions shown
by (10) are identical to that shown by (9), as a result, we can said the both
filters are adjoints. The SPICE simulation results of the CM GmC filter are
shown in Fig. 8, whose behavior is the same like that shown
in Fig. 6.
SYNTHESIS OF VM AND CM GmC FILTERS USING CMOS OTAs AND CURRENT CONVEYORS
The filters shown in Fig. 5 and 7 are adjoints.
Although their symbolic transfer functions are identical, their frequency response
can vary depending on the design of the transconductor to accomplish the desired
value of Gm of each OTA. In this manner, the goal of an analog IC designer is
to guarantee that the CMOS OTA or current conveyor perform the behavior of the
Gm needed to the physical implementation of the adjoint filters. For instance,
in SánchezSinencio and SilvaMartínez (2000) is shown how to
design CMOS OTAs and its noise characterization is described in SánchezLópez
and TleloCuautle (2006). However, nowadays there is a greatest interest to
use novel active devices to synthesize transconductors, such devices are known
as Current Conveyors (CCs). Torres and Tlelo (2004) introduced the design of
transconductors using two second generation CCs. Masmoudi et al. (2005)
demonstrated its usefulness to design active filters. Salem et al. (2006)
proved that CCs can reach high frequencies and Fakhfakh et al. (2007)
introduced a symbolic procedure for noise characterization of this devices.
An analog IC designer has a tradeoff to choose OTAs or CCs to design GmC filters. The authors recommend to use CCs because they facilitate the implementation of transconductors with multiple outputcurrents and because its design is based on the interconnection of voltage followers with either current followers or current mirrors. Elsewhere, in Torres and Tlelo (2004) is shown the design of CMOS CCs to synthesize a transconductor to implement Gm.
CONCLUSION
It has been introduced a method to transform VM opampRC filters to CM GmC filters. The method was applied to a secondorder lowfrequency opampRC filter with stable high Q.
The transformation process began by transforming a Ynetwork embedded in the original opampRC circuit, to a Δ network to eliminate one floating node. Second, three nodal equations were formulated from the opampRC circuit and the equations were arranged to have a general form suitable for synthesis purposes. Third, each equation was synthesized using OTAC blocks to obtain a VM GmC filter with all capacitors connected to ground. SPICE simulation results were performed to show that the transformed VM GmC filter has the same frequency behavior than the opampRC filter. Further, the symbolic transfer functions were derived.
It were described the rules to transform a VM GmC filter to a CM GmC filter.
SPICE simulations were performed to show that the adjoint CM GmC filter has
the same frequency behavior that the original VM GmC filter. The symbolic transfer
functions were derived to show that the VM ones are the same that the CM ones,
so that one can conclude that both filters are adjoints and that they perform
the same behavior than the original opampRC filter.
Finally, some guidelines to design the transconductors were briefly described to highlight the usefulness of current conveyors to design highfrequency active filters.
ACKNOWLEDGMENT
This work has been supported by CONACyT/México under the project number 48396Y.