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nth-order current transfer function synthesis using current differencing buffered amplifier: signal-flow graph approach
Microelectronics Journal Microelectronics Journal 31 (2000) 49–53 www.elsevier.com/locate/mejo
nth-order current transfer function synthesis using current differencing buffered amplifier: signal-flow graph approach ¨ zogˇuz C. Acar*, S. O Istanbul Technical University, Faculty of Electrical-Electronics Engineering, 80626 Maslak, Istanbul, Turkey Accepted 12 April 1999
Abstract The authors present a general synthesis method for the realisation of nth-order current transfer function using current differencing buffered amplifier which can easily be derived from a commercially available active component, AD844. The use of this active component simplifies the implementation of analogue signal processing filters and hence makes the proposed circuit attractive. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Current-mode circuits; Current differencing buffered amplifiers; Signal-flow graphs
1. Introduction Current-mode (CM) circuits are receiving much attention for their potential advantages such as inherent wider bandwidth, simpler circuitry, lower power consumption and wider dynamic range [1]. In the last decade, owing to these advantages of the CM operation, several CM current conveyor (CC)-based filter realisations have been presented (see Refs. [2,3] and the references cited therein). The purpose of this study is to propose a new method for the realisation of the most general nth-order current transfer function using a new active element, namely current differencing buffered amplifier (CDBA) [4] in order to simplify the design of current-mode filters widely used in analogue signal processing systems. First, the active element CDBA and its equivalent circuit involving dependent sources are given. Also, an implementation of CDBA is included using commercially available current feedback amplifier (CFA), AD844 [5,6]. Finally, the realisation procedure is derived by the use of signal-flow graph corresponding to given transfer function.
2. Circuit description The circuit symbol of the proposed active element, CDBA, is shown in Fig. 1(a), where p and n are input * Corresponding author. Fax: 1 90-212-285-3679. E-mail addresses: [email protected] (C. Acar); [email protected] ¨ zogˇuz) (S. O
terminals and w and z are output terminals. This element is equivalent to the circuit in Fig. 1(b), which involves dependent current and voltage sources. Its characteristics can be modeled as 32 3 2 3 2 vz 0 0 1 21 iz 76 7 6 7 6 6 v 7 6 1 0 0 0 76 i 7 76 w 7 6 w7 6 76 7: 6 76 1 76 7 6 7 6 6 v p 7 6 0 0 0 0 7 6 ip 7 54 5 4 5 4 vn in 0 0 0 0 According to the equations described earlier and equivalent circuit of Fig. 1(b), current through z-terminal follows the difference of the currents through p-terminal and n-terminal, and hence we name z-terminal as current differencing output. We also name p-terminal as positive (non-inverting) input and n-terminal as negative (inverting) input. Moreover, voltage of w-terminal follows the voltage of z-terminal. Hence, we name w-terminal as voltage output. Finally, note that the input terminals p and n, are internally grounded. Note that the difference of the input currents are converted into the output voltage ‘w’ through the two-terminal element connected at the z-terminal. Therefore, this element can be considered as a transimpedance amplifier and it is similar to CFA. A possible implementation of CDBA using commercially available CFA, AD844 is given in Fig. 1(c).
3. Realisation procedure
0026-2692/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0026-269 2(99)00088-9
Let the nth-order current transfer function be expressed
50
¨ zogˇuz / Microelectronics Journal 31 (1999) 49–53 C. Acar, S. O
Fig. 1. (a) Symbol of CDBA, (b) equivalent circuit of CDBA and (c) implementation of CDBA using CFAs.
as T s
Iout s a sn 1 an21 sn21 1 … 1 a1 s 1 a0 nn Iin s s 1 bn21 sn21 1 … 1 b1 s 1 b0
2
where Iin and Iout are the input and output currents. Numerator is a polynomial with positive and negative real coefficients. Denominator is a Hurwitz polynomial with positive real coefficients. This function can be represented by the signal-flow graph of Fig. 2, whose node
signals are assumed to be currents. Using well-known Mason gain formula, it is possible to verify that the graph transfer function from the input node 0 to the output node n 1 1 is equal to T(s). Now, using this signal-flow graph, the nth-order current transfer function T(s) can be realised by the active-RC circuit involving CDBAs. For this realisation, it is sufficient to know the active subcircuits corresponding to the subgraphs, which consist of the branches outgoing from each node. The subgraphs consisting of the outgoing
Fig. 2. Signal-flow graph model realising T(s) of Eq. (2).
¨ zogˇuz / Microelectronics Journal 31 (1999) 49–53 C. Acar, S. O
51
Fig. 3. The subgraphs and their corresponding active subcircuits involving CDBAs: (a) current distributor and (b) current integrator.
branches and their proposed active-RC realisations using CDBAs are shown for nodes i 0 and i n 1 1 in Fig. 3(a), and for i 1, 2,…, n in Fig. 3(b). For the subcircuit in Fig. 3(a), output signals are related to
the input signal as follows: > I 0j tj21 Ii ; j 1; 2; …; k
3
where tj21 aj21 and k n 1 1 for i 0, and tj21 bj21 and
Fig. 4. Third-order allpass filter realising T(s) of Eq. (5).
¨ zogˇuz / Microelectronics Journal 31 (1999) 49–53 C. Acar, S. O
52
Fig. 5. Experimental results for the third-order allpass filter in Fig. 4. —: Ideal; W: measured.
k n for i n 1 1. Note that Ij and Ij0 are different currents. Ij is the jth node current in Fig. 2 and Ij0 is the current incoming to the node j by the branch whose transmittance is tj21. For the subcircuit in Fig. 3(b), input–output relation can be given as follows: I 0i11
1 I; s i
i 1; 2; …; n:
4
where I 0 i11 is also the current incoming to the node i 1 1 by the branch in Fig. 2 whose transmittance is 1/s. In fact, the first subcircuit in Fig. 3(a) is a current distributor and that of Fig. 3(b) is a simple current integrator which are composed of CDBAs. One can readily obtain the CDBA-RC circuit by interconnecting these corresponding subcircuits according to the overall signal-flow graph of Fig. 2. Note that the input and the outputs of the proposed subcircuits are grounded. Therefore, all these subcircuits can be interconnected without using any additional interconnection networks. In Fig. 3(a), the polarity of the transmittance tj is realised after the interconnections of the subcircuits. Namely, if tj is positive, the floating terminal of the corresponding resistor (i.e. Gi( j11) in Fig. 3(a)) is connected to the non-inverting terminal of the CDBA in the related succeeding stage and the conductance of this resistor takes the value of tj (i.e. Gi( j11) tj). In the same way, if tj is negative, the floating terminal of the corresponding resistor is connected to the inverting terminal of the succeeding CDBA and this resistor takes the value of 2 tj (i.e. Gi( j11) 2 tj). As an application of this procedure, the third-order
allpass transfer function, T s
Iout a s3 1 a2 s2 1 a1 s 1 a0 3 3 Iin b3 s 1 b2 s2 1 b1 s 1 b0 2s3 1 b2 s2 2 b1 s 1 b0 s3 1 b2 s2 1 b1 s 1 b0
5
is realised as an example, and the circuit thus obtained is shown in Fig. 4 where the element values denote conductances and capacitances. Note from Fig. 4 that all the resistor values can be obtained in terms of the coefficients of the transfer function, therefore the design of the circuit is very simple. However, as the order of the filter increases, this may cause to increase the spread of the passive components values, in which case, it is always possible to optimise not only the spread of the passive elements but also the dynamic range of the obtained circuit [7] by scaling branch transmittances in the signalflow graph of Fig. 2 in such a way that loop-gains and forward path-gains remain unchanged, as this type of scaling introduces n independent parameters to the design. 4. Experimental results In order to verify the theoretical analysis, the circuit in Fig. 4 is built to realise third-order allpass filter with a Butterworth polynomial using commercially available CFAs, AD844s. In order to get a cut-off frequency of 15.9 kHz, the passive element values have been chosen as follows: R02 R03 R42 R43 5 kV, and R01 R04
¨ zogˇuz / Microelectronics Journal 31 (1999) 49–53 C. Acar, S. O
R41 R12 R23 R34 R0 R4 10 kV. All the capacitor values are taken as 1 nF. The experimental results verifying theoretical analysis are shown in Fig. 5. 5. Conclusion A simple synthesis procedure has been presented for generating not only the most general nth-order but also 2nd-order CDBA-based current-mode filters. The method presented here is straightforward and simple in that it gives not only the configuration but also the element values directly from the coefficients of the nth-order transfer function. As seen from the example, only n capacitors (all of them being equal) and n 1 2 commercially implementable active components (i.e. CDBAs) are required for the realisation. The proposed filter circuit uses at most 3n 1 3 resistors. However, this number depends on non-zero coefficients, and hence it is considerably reduced in realisation of lowpass, bandpass, highpass and allpole filter responses. The use of CDBA as active component simplifies the implementation. The effects of the parasitic input impedances of the CDBAs on filter performances can be reduced by selecting the impedance-scaling factor properly as stated by Svoboda [8]. Note that designer can scale the overall signal-flow graph of Fig. 2 so that corresponding transfer function is not changed. This feature of the signal-flow graphs gives flexibility to the design. As the CDBA is equivalent to the circuit composed of only two CFAs, the proposed circuit is comparable with CFA-based nth-order filter circuit in Ref. [9]. Finally, in this work, inverse-follow-the-leader-feedback
53
(IFLF) topology in Fig. 2 is chosen as it has much lower component sensitivities [10,11] than the cascade approach. However, another CDBA-based current-mode filter circuit different from that proposed in this work can be derived in a similar way using another topology different than IFLF.
References [1] G.W. Roberts, A.S. Sedra, All current-mode frequency selective circuits, Electron. Lett. 25 (1989) 759–761. [2] S.I. Liu, H.W. Tsao, J. Wu, Cascadable current-mode single CCIIbiquad, Electron. Lett. 26 (1990) 2005–2006. [3] R. Senani, A simple approach of deriving single-input-multipleoutput current-mode biquad filters, Frequenz 50 (1996) 124–127. ¨ zogˇuz, A new versatile building block: current differen[4] C. Acar, S. O cing buffered amplifier suitable for analog signal-processing filters, Microelectronics J. 30 (1999) 157–160. [5] Analog Devices, Linear Products Data Book, Norwood, MA, 1990. [6] J.A. Svoboda, L. Mcgory, S. Webb, Applications of a commercially available current conveyor, Int. J. Electron. 70 (1991) 159–164. [7] C. Acar, H. Kuntman, Limitations on input signal level in currentmode active-RC filters using CCIIs, Electron Lett. 32 (1996) 1461– 1462. [8] J.A. Svoboda, Transfer function synthesis using current conveyors, Int. J. Electronics 76 (1994) 611–614. [9] C. Acar, High-order voltage transfer function synthesis using commercially available active component, AD844: signal-flow graph approach, Electron Lett. 32 (1996) 1933–1934. [10] K.R. Laker, R. Schaumann, M.S. Ghausi, Multiple-loop feedback topologies for the design of low-sensitivity active filters, IEEE Trans. Circ. Syst. CAS-26 (1979) 1–20. [11] D.H. Chiang, R. Schaumann, A CMOS fully-balanced continuoustime IFLF filter design for read/write channels, Proc. IEEE Int. Symp. On Circuits and Systems 1 (1990) 1.176–1.170.
nth-order current transfer function synthesis using current differencing buffered amplifier: signal-flow graph approach ¨ zogˇuz C. Acar*, S. O Istanbul Technical University, Faculty of Electrical-Electronics Engineering, 80626 Maslak, Istanbul, Turkey Accepted 12 April 1999
Abstract The authors present a general synthesis method for the realisation of nth-order current transfer function using current differencing buffered amplifier which can easily be derived from a commercially available active component, AD844. The use of this active component simplifies the implementation of analogue signal processing filters and hence makes the proposed circuit attractive. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Current-mode circuits; Current differencing buffered amplifiers; Signal-flow graphs
1. Introduction Current-mode (CM) circuits are receiving much attention for their potential advantages such as inherent wider bandwidth, simpler circuitry, lower power consumption and wider dynamic range [1]. In the last decade, owing to these advantages of the CM operation, several CM current conveyor (CC)-based filter realisations have been presented (see Refs. [2,3] and the references cited therein). The purpose of this study is to propose a new method for the realisation of the most general nth-order current transfer function using a new active element, namely current differencing buffered amplifier (CDBA) [4] in order to simplify the design of current-mode filters widely used in analogue signal processing systems. First, the active element CDBA and its equivalent circuit involving dependent sources are given. Also, an implementation of CDBA is included using commercially available current feedback amplifier (CFA), AD844 [5,6]. Finally, the realisation procedure is derived by the use of signal-flow graph corresponding to given transfer function.
2. Circuit description The circuit symbol of the proposed active element, CDBA, is shown in Fig. 1(a), where p and n are input * Corresponding author. Fax: 1 90-212-285-3679. E-mail addresses: [email protected] (C. Acar); [email protected] ¨ zogˇuz) (S. O
terminals and w and z are output terminals. This element is equivalent to the circuit in Fig. 1(b), which involves dependent current and voltage sources. Its characteristics can be modeled as 32 3 2 3 2 vz 0 0 1 21 iz 76 7 6 7 6 6 v 7 6 1 0 0 0 76 i 7 76 w 7 6 w7 6 76 7: 6 76 1 76 7 6 7 6 6 v p 7 6 0 0 0 0 7 6 ip 7 54 5 4 5 4 vn in 0 0 0 0 According to the equations described earlier and equivalent circuit of Fig. 1(b), current through z-terminal follows the difference of the currents through p-terminal and n-terminal, and hence we name z-terminal as current differencing output. We also name p-terminal as positive (non-inverting) input and n-terminal as negative (inverting) input. Moreover, voltage of w-terminal follows the voltage of z-terminal. Hence, we name w-terminal as voltage output. Finally, note that the input terminals p and n, are internally grounded. Note that the difference of the input currents are converted into the output voltage ‘w’ through the two-terminal element connected at the z-terminal. Therefore, this element can be considered as a transimpedance amplifier and it is similar to CFA. A possible implementation of CDBA using commercially available CFA, AD844 is given in Fig. 1(c).
3. Realisation procedure
0026-2692/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0026-269 2(99)00088-9
Let the nth-order current transfer function be expressed
50
¨ zogˇuz / Microelectronics Journal 31 (1999) 49–53 C. Acar, S. O
Fig. 1. (a) Symbol of CDBA, (b) equivalent circuit of CDBA and (c) implementation of CDBA using CFAs.
as T s
Iout s a sn 1 an21 sn21 1 … 1 a1 s 1 a0 nn Iin s s 1 bn21 sn21 1 … 1 b1 s 1 b0
2
where Iin and Iout are the input and output currents. Numerator is a polynomial with positive and negative real coefficients. Denominator is a Hurwitz polynomial with positive real coefficients. This function can be represented by the signal-flow graph of Fig. 2, whose node
signals are assumed to be currents. Using well-known Mason gain formula, it is possible to verify that the graph transfer function from the input node 0 to the output node n 1 1 is equal to T(s). Now, using this signal-flow graph, the nth-order current transfer function T(s) can be realised by the active-RC circuit involving CDBAs. For this realisation, it is sufficient to know the active subcircuits corresponding to the subgraphs, which consist of the branches outgoing from each node. The subgraphs consisting of the outgoing
Fig. 2. Signal-flow graph model realising T(s) of Eq. (2).
¨ zogˇuz / Microelectronics Journal 31 (1999) 49–53 C. Acar, S. O
51
Fig. 3. The subgraphs and their corresponding active subcircuits involving CDBAs: (a) current distributor and (b) current integrator.
branches and their proposed active-RC realisations using CDBAs are shown for nodes i 0 and i n 1 1 in Fig. 3(a), and for i 1, 2,…, n in Fig. 3(b). For the subcircuit in Fig. 3(a), output signals are related to
the input signal as follows: > I 0j tj21 Ii ; j 1; 2; …; k
3
where tj21 aj21 and k n 1 1 for i 0, and tj21 bj21 and
Fig. 4. Third-order allpass filter realising T(s) of Eq. (5).
¨ zogˇuz / Microelectronics Journal 31 (1999) 49–53 C. Acar, S. O
52
Fig. 5. Experimental results for the third-order allpass filter in Fig. 4. —: Ideal; W: measured.
k n for i n 1 1. Note that Ij and Ij0 are different currents. Ij is the jth node current in Fig. 2 and Ij0 is the current incoming to the node j by the branch whose transmittance is tj21. For the subcircuit in Fig. 3(b), input–output relation can be given as follows: I 0i11
1 I; s i
i 1; 2; …; n:
4
where I 0 i11 is also the current incoming to the node i 1 1 by the branch in Fig. 2 whose transmittance is 1/s. In fact, the first subcircuit in Fig. 3(a) is a current distributor and that of Fig. 3(b) is a simple current integrator which are composed of CDBAs. One can readily obtain the CDBA-RC circuit by interconnecting these corresponding subcircuits according to the overall signal-flow graph of Fig. 2. Note that the input and the outputs of the proposed subcircuits are grounded. Therefore, all these subcircuits can be interconnected without using any additional interconnection networks. In Fig. 3(a), the polarity of the transmittance tj is realised after the interconnections of the subcircuits. Namely, if tj is positive, the floating terminal of the corresponding resistor (i.e. Gi( j11) in Fig. 3(a)) is connected to the non-inverting terminal of the CDBA in the related succeeding stage and the conductance of this resistor takes the value of tj (i.e. Gi( j11) tj). In the same way, if tj is negative, the floating terminal of the corresponding resistor is connected to the inverting terminal of the succeeding CDBA and this resistor takes the value of 2 tj (i.e. Gi( j11) 2 tj). As an application of this procedure, the third-order
allpass transfer function, T s
Iout a s3 1 a2 s2 1 a1 s 1 a0 3 3 Iin b3 s 1 b2 s2 1 b1 s 1 b0 2s3 1 b2 s2 2 b1 s 1 b0 s3 1 b2 s2 1 b1 s 1 b0
5
is realised as an example, and the circuit thus obtained is shown in Fig. 4 where the element values denote conductances and capacitances. Note from Fig. 4 that all the resistor values can be obtained in terms of the coefficients of the transfer function, therefore the design of the circuit is very simple. However, as the order of the filter increases, this may cause to increase the spread of the passive components values, in which case, it is always possible to optimise not only the spread of the passive elements but also the dynamic range of the obtained circuit [7] by scaling branch transmittances in the signalflow graph of Fig. 2 in such a way that loop-gains and forward path-gains remain unchanged, as this type of scaling introduces n independent parameters to the design. 4. Experimental results In order to verify the theoretical analysis, the circuit in Fig. 4 is built to realise third-order allpass filter with a Butterworth polynomial using commercially available CFAs, AD844s. In order to get a cut-off frequency of 15.9 kHz, the passive element values have been chosen as follows: R02 R03 R42 R43 5 kV, and R01 R04
¨ zogˇuz / Microelectronics Journal 31 (1999) 49–53 C. Acar, S. O
R41 R12 R23 R34 R0 R4 10 kV. All the capacitor values are taken as 1 nF. The experimental results verifying theoretical analysis are shown in Fig. 5. 5. Conclusion A simple synthesis procedure has been presented for generating not only the most general nth-order but also 2nd-order CDBA-based current-mode filters. The method presented here is straightforward and simple in that it gives not only the configuration but also the element values directly from the coefficients of the nth-order transfer function. As seen from the example, only n capacitors (all of them being equal) and n 1 2 commercially implementable active components (i.e. CDBAs) are required for the realisation. The proposed filter circuit uses at most 3n 1 3 resistors. However, this number depends on non-zero coefficients, and hence it is considerably reduced in realisation of lowpass, bandpass, highpass and allpole filter responses. The use of CDBA as active component simplifies the implementation. The effects of the parasitic input impedances of the CDBAs on filter performances can be reduced by selecting the impedance-scaling factor properly as stated by Svoboda [8]. Note that designer can scale the overall signal-flow graph of Fig. 2 so that corresponding transfer function is not changed. This feature of the signal-flow graphs gives flexibility to the design. As the CDBA is equivalent to the circuit composed of only two CFAs, the proposed circuit is comparable with CFA-based nth-order filter circuit in Ref. [9]. Finally, in this work, inverse-follow-the-leader-feedback
53
(IFLF) topology in Fig. 2 is chosen as it has much lower component sensitivities [10,11] than the cascade approach. However, another CDBA-based current-mode filter circuit different from that proposed in this work can be derived in a similar way using another topology different than IFLF.
References [1] G.W. Roberts, A.S. Sedra, All current-mode frequency selective circuits, Electron. Lett. 25 (1989) 759–761. [2] S.I. Liu, H.W. Tsao, J. Wu, Cascadable current-mode single CCIIbiquad, Electron. Lett. 26 (1990) 2005–2006. [3] R. Senani, A simple approach of deriving single-input-multipleoutput current-mode biquad filters, Frequenz 50 (1996) 124–127. ¨ zogˇuz, A new versatile building block: current differen[4] C. Acar, S. O cing buffered amplifier suitable for analog signal-processing filters, Microelectronics J. 30 (1999) 157–160. [5] Analog Devices, Linear Products Data Book, Norwood, MA, 1990. [6] J.A. Svoboda, L. Mcgory, S. Webb, Applications of a commercially available current conveyor, Int. J. Electron. 70 (1991) 159–164. [7] C. Acar, H. Kuntman, Limitations on input signal level in currentmode active-RC filters using CCIIs, Electron Lett. 32 (1996) 1461– 1462. [8] J.A. Svoboda, Transfer function synthesis using current conveyors, Int. J. Electronics 76 (1994) 611–614. [9] C. Acar, High-order voltage transfer function synthesis using commercially available active component, AD844: signal-flow graph approach, Electron Lett. 32 (1996) 1933–1934. [10] K.R. Laker, R. Schaumann, M.S. Ghausi, Multiple-loop feedback topologies for the design of low-sensitivity active filters, IEEE Trans. Circ. Syst. CAS-26 (1979) 1–20. [11] D.H. Chiang, R. Schaumann, A CMOS fully-balanced continuoustime IFLF filter design for read/write channels, Proc. IEEE Int. Symp. On Circuits and Systems 1 (1990) 1.176–1.170.