Systematic Synthesis of Compressive VDTA-Based T-T Filter with Orthogonal Control of fo and Q

Abstract—Based on sixteen nullor-mirror models of the voltage differencing transconductance amplifier (VDTA) and port admittance matrices of the tow-Thomas (T-T) filter with orthogonal control between the characteristic frequency(fo) and figure of merit (Q),two different categories of the voltage-mode and transconductance-mode T-T filters are synthesized by the means of the nodal admittance matrix (NAM) expansion method.The category A filter that employs two compressive VDTAs and two grounded capacitors includes four structures,and the category B filter that uses two compressive VDTAs,two grounded capacitors,and one grounded resistor,also includes four structures.These circuits are suitable for integrated circuit manufacture,and their parameters fo and Q can be orthogonally adjusted with varying the bias currents of VDTAs.After the paper and pencil test is completed,the computer analyses,including alternating current (AC),parameter sweep,Monte Carlo (MC),and noise analyses,are performed to support the synthesis approach.

1.Introduction

A voltage differencing transconductance amplifier (VDTA) was initially introduced by Biolek in 2008[1].Compared with the operational transconductance amplifier (OTA),VDTA has two instructive advantages.One is that it is highly suitable for electronic adjusting circuit design due to its two tunable transconductance parameters,and the other is its compressive structures,which can be easily realized in some applications.Therefore,VDTA is a fundamental,general,and versatile transconductance-mode device.A mountain of analog circuits based on VDTA,like oscillators,filters,grounded and floating inductances,and floating frequency-dependent negative resistance(FDNR),have been designed[2]-[11].Unfortunately,most of the design methods,such as the operation simulation method,element simulation method,and signal-flow graph method,are of ordinary circuit-theory techniques for the circuits using VDTAs.They lack systematic characteristics.Until now,the approach to systematic synthesis for the filter employing VDTAs has not been studied,which is worthy of further research.

On the other hand,a systematic method to synthesize the linear active circuit,namely the nodal admittance matrix (NAM) expansion,was put forward by Haighet al.in 2006[12],and applied to the second generation current conveyor (CCII) and second generation inverting current conveyor (ICCII) by Saad and Soliman in 2008[13].Until now,this approach has been extended to OTA,the current-controlled current conveyor transconductance amplifier(CCCCTA),and second generation current-controlled conveyor (CCCII) to achieve the synthesis for analog circuits like the gyrator,oscillator,and filter[12]-[23].Consequently,in advanced analog circuit synthesis,it is significant to synthesize various circuits using VDTAs by the NAM expansion method.In 2018,the sixteen nullor-mirror models of VDTA were proposed and applied to synthesize VDTA-based Wien oscillators by our team in [24].

The objective of the paper is to utilize the NAM expansion method to synthesize the tow-Thomas (T-T) filter employing VDTAs.First,according to the nullor-mirror models of VDTA and port admittance matrices of the T-T filter with orthogonal control between the characteristic frequency (fo) and figure of merit (Q),and utilizing the NAM expansion method,eight different configurations of the voltage-mode and transconductance-mode T-T filters are synthesized,employing two compressive VDTAs,two grounded capacitors,or two grounded capacitors and one grounded resistor.Since employing less active elements and using grounded capacitors,the circuits are applicable for integrated circuit manufacture.And the filters also provide good features,namely linear,orthogonal,and electronic control between the characteristic frequency and figure of merit.Eventually,the validity of one of the synthesized circuits is proved by hand analyses and computer simulation.

2.Basis of Circuit Synthesis

2.1.NAM of T-T Filters with Orthogonal Control of fo and Q

The original T-T filter has been reported in [23].Recently,[20]gave a modified T-T filter with orthogonal control offoandQ,whose port admittance matrices are as follows:

In addition,foandQof the filter is,respectively

Under the conditions ofC1=C2=CandG1=G2=G,the above formula would be reduced to

Equation (3) shows that the circuit provides a good feature:Orthogonal control betweenfoandQ.It can also be seen that the T-T filter possesses four original admittance matrices,providing the foundation of systematic synthesis of the VDTA-based T-T filter.

2.2.Basic Concept of VDTA

Fig.1 (a) shows the symbol of VDTA,Fig.1 (b) gives the bipolar junction transistor (BJT) realization of compressive VDTA,and (4) presents its terminal relations[3],[6].

Fig.1.Circuit symbol and BJT realization for compressive VDTA:(a) circuit symbol and (b) possible BJT realization.

For VDTA,to implement with the BJT technology,the two transconductance gains of the compressive VDTA are expressed in (5),whereIB1andIB2represent bias currents of the first and second VDTA,respectively;VTrepresents the usual thermal voltage;gm1and gm2are

3.Systematic Synthesis for T-T Filter

3.1.Synthesis for Category A

Our purpose is to synthesize the T-T filter with orthogonal control betweenfoandQ.First,imposeC1=C2=CandG1=G2=G.Next,in the light of the NAM expansion approach,starting from (1a),and noting that to contain seven nodes in the filter,the first step for the expansion of NAM is adding four blank rows and columns,and then utilizing the first nullator to connect columns 3 and 4 for shifting–G4to the position (2,4).A first current mirror is inserted between rows 2 and 4,thereby replacing–G4to beG4at the positions (4,4).The second nullator is then inserted between columns 1 and 5,moving theGto the position (2,5).The first norator links rows 2 and 5 to moveGto the position (5,5).The third nullator links columns 2 and 6,moving–GandGfrom the second column to the sixth column.The second current mirror connects rows 1 and 6 to replace–Gto beGat the position (6,6).The third current mirror is then inserted between columns 3 and 7,movingGto be–Gat the position (7,6).At last,the fourth nullator is inserted between the seventh column and grounded to recover the losing–GandGon the seventh column.The extension NAM matrix to contain pathological elements constituted with the bracket notation is depicted in (6):

In (6),G3,G4,andGindicate the conductances of nodes 3,4,5 to the ground,respectively.Also,the conductance between nodes 6 and 7 is a floating conductance.As (6) shows,this extension matrix includes four different pairs of nullor-mirror elements,two grounded capacitors,three grounded conductances,and one floating conductance.

After addingC1andC2at nodes 1 and 2,respectively,the equivalent pathological circuit drawn by (6) is displayed in Fig.2.Employing the pathological models for VDTA and Fig.2,two equivalent compressive VDTAbased circuits can be accomplished,as given in Figs.3 (a) and (b).

Fig.2.Pathological equivalent model displayed with (6).

It is noteworthy that in Fig.3,G=gm11=gm12=IB11/2VT=IB12/VT,G3=gm21=IB21/2VT,andG4=gm22=IB22/2VT.

It can be foreseen that starting from (1) and practicing various conceivable combinations of the lent pathological elements would propagate sixty-four different configurations of the expanded matrixes and equivalent pathological models.Nevertheless,only four equivalent circuits based on the compressive VDTA,given in Fig.3,are synthesized.Consequently,it can be found that the category A filter,employing two compressive VDTA and two grounded capacitors,has four different configurations in all.

3.2.Synthesis for Category B

The synthesis for the category B filter proceeds in almost the same way for the category A filter.Initiating from(1a),underC1=C2=CandG1=G2=G,and through a series of NAM expansion procedures,it has been found that the matrix for expanding is (7),the pathological equivalent models are Fig.4,and that equivalent VDTA-based circuits are shown in Fig.5,whereG=gm11=gm12=gm21=IB11/2VT=IB12/VT=IB21/VTandG4=gm22=IB22/2VT.

Fig.3.Four equivalent circuits from Fig.2:(a) topology I,(b) topology II,(c) topology III,and (d) topology IV.

Also,using (1) and the same synthesis would propagate sixty-four different configurations of expanded matrixes and equivalent pathological models.But only four equivalent compressive VDTA-based circuits are synthesized as shown in Fig.5.Therefore,it can be found that the category B filter has four different configurations in all,which employs two compressive VDTA,two grounded capacitors,and one grounded conductance.

Fig.4.Pathological equivalent model displayed with (7).

4.Circuit Analyses

It is worthy mentioning that the circuits synthesized in Sections 2 and 3 are filter loops rather than filters themselves.Now,let us consider the filter loop in Fig.3 (a).If thenterminal of VDTA1 is lifted off ground and driven byVi,whileIo1,Io2,andIo3are served as current outputs,andV1,V2,andV3are served as voltage outputs,then the T-T filter with two compressive VDTAs,which realizes the voltage mode and transconductance mode,can be obtained,as shown in Fig.6.

Fig.5.Four equivalent circuits from Fig.4:(a) topology I,(b) topology II,(c) topology III,and (d) topology IV.

This circuit includes a global loop and a local feedback loop,whose gains areG2/s2C2and–GG4/sCG3,respectively,wheresis the complex frequency.Applying Mason’s formula,the graph determinant of the filter is written as

Fig.6.T-T filter based on Fig.3 (a).

According to Fig.6,the forward transfers fromVitoV1,toV2,and toV3can be written as −G/sC,G2/s2C2+GG4/sCG3,andG/G3,respectively.The equivalent voltage-mode transfer functions can be readily written as

where

The forward transfers fromVitoIo1,toIo2,and toIo3are –G2/sC,G,and −GG4/G3,respectively.The corresponding transconductance-mode transfer functions can be readily calculated as

Obviously,not only a band-pass filtering function is available in this circuit,but also two high-pass filtering functions,which are not achieved in the original T-T filter.However,the low-pass transfer function is missed.Combining (3) with (5),the characteristic frequencyfoand figure of meritQare acquired,respectively

whereIB=IB11=IB12.Equation (12) states thatfocan be linearly tuned by adjusting the bias currentIBand thatQcan be independently and linearly controlled by adjusting the bias currentIB21without the influence onfo.

Fig.7.Alternative circuit based on Fig.3 (a).

If thepinput of VDTA2 is lifted off ground and driven withVi,the rest is as before,the obtained T-T filter is described in Fig.7.

The graph determinant of the circuit is similar to (8).From Fig.7,forward transfers fromVitoV1and toV2areGG4/s2C2andG4/sC,respectively.The equivalent voltage-mode transfer functions are as follows:

The forward transfers fromVitoIo1and toIo2areG4G2/s2C2and −GG4/sC,respectively.The equivalent transconductance-mode transfer functions can be readily calculated as

ConsideringIo2=−G3(Vi−V3) andIo3=−G4V3,and combining (14) yield the following transfer functions:

It can be seen thatfoandQare the same as (12).Obviously,not only the low-pass and band-pass transfer functions are available in this circuit,but also the band-stop transfer functions that are not achieved in the original T-T filter.

According to (2) and the transconductance gain equation isG=IB/2VT.all passive and active sensitivities of the synthesized filter are shown in (16):

These sensitivities are fairly low and similar to those of a passive resistor-capacitor (RC) filter with the same responses.

For other circuits,the analysis process is omitted to limit the paper length.

5.Simulation Analyses

A computer simulation was carried out based on the circuit in Fig.7,where compressive VDTA was built through the models of transistors PR200N and NR200N.WhenC=1 nF,IB=327 µA,andIB21=IB22=327 µA,the design parameters arefo=1 MHz,Q=1,andHPB=6.28 mS,whereHPBis the passband gain.The AC analysis results of the voltage-mode T-T filter are given in Fig.8,which shows that the actual characteristic frequencyfoa=981 kHz and the actual figure of meritQa=1.01,so the deviation forfois −1.9% and that forQis 1.0%.The AC analysis results of the transconductance-mode T-T filter are showcased in Fig.9,which shows thatfoaandQaare the same asfoandQ,andHPBa=6.07 mS,so the deviation forHPBis −3.3%,whereHPBais the actual passband gain.

Fig.8.Frequency responses of voltage-mode circuit in Fig.7.

Fig.9.Frequency responses of transconductance-mode circuit in Fig.7.

To demonstrate the control feature ofQby adjustingIB21,letIB=327 µA andIB22=82 µA.WhenIB21is 41 µA,82 µA,164 µA,and 327 µA,the design value for the figure of meritQis 0.5,1.0,2.0,and 4.0.Results of the parameter sweep analysis are shown in Fig.10.

To demonstrate the control feature offoby adjustingIB,IB21andIB22are kept the same as before.WhenIBis 327 µA,164 µA,and 32.7 µA,the design value forfois 1 MHz,500 kHz,and 100 kHz.The parameter sweep analysis results are shown in Fig.11.

Fig.10.Frequency responses of transconductance-mode band-pass filter controlled by IB21 but keeping fo=1 MHz.

Fig.11.Frequency responses of transconductance-mode band-pass filter controlled by IB but keeping Q=1.0.

The AC analysis and Monte Carlo (MC) analysis results ofIo2near the pole frequencies for the circuit of Fig.5 are given by Fig.12,whereIB=IB21=IB22=327 µA,the values ofC1,C2,IB,IB21,IB22,and area of BJTs are with 5% tolerance and in the Gaussian distribution,which gives little differences.

At last,the noise analysis of the circuit in Fig.7 withIB=IB21=IB22=327 µA is simulated,as depicted in Fig.13,which shows that the noise density value is much small.

It is evident that the computer analyses results concur with the synthesis method.

Fig.12.AC and Monte Carlo analyses of circuit in Fig.7.

6.Conclusions

This paper utilizes the NAM expansion method to obtain a family of T-T filters using compressive VDTAs,namely eight voltage-mode/transconductance-mode T-T filters.The synthesized double mode circuits contain many advantages,such as the use of grounded capacitors,compressive structures,low sensitivities,and electronic orthogonal control offoandQ.Also,the synthesized transconductance-mode T-T filter has no loading at the input and output terminals.The results through hand and computer analyses have verified the involved theory.

Fig.13.Output noise spectral density of Fig.7.

Acknowledgment

The author would like to thank the anonymous reviewers for their suggestions.