96 T. DOSTÁL, V. AMA, BIUADS BASED O SIGLE EGATIVE IMEDAE OVETE Biquads Based on Single egative Impedance onverter Implemented by lassical urrent onveyor Tomáš DOSTÁL, Vladimír AMA Dept. of adio Electronics, Brno University of Technology, urkyňova 8, 6 00 Brno, zech epublic dostal@feec.vutbr.cz Abstract. The paper deals with continuous-time A biquadratic (second-order) filters based on the single classical three-port current conveyors, so called second generation II. A unique application of the II is discussed, when the terminals Y and are connected together. Using this way a negative impedance converter is obtained as a suitable building block in synthesis of several biquads, including notch filters. Keywords Analogue circuits, active filters, biquads, current conveyors, negative impedance converters.. Introduction It is well known that precisely tailored frequency filters can be produced with standard operational amplifiers [8], but for audio application only. In higher frequency range it is better to use some of modern active functional blocks. Such versatile and powerful building blocks are the current conveyors () [6]. ontinuous-time active filters based on the have recently found attractive considerable attention. This stems from inherent advantages of the circuits, namely low supply voltages and power, current operational mode possibility, well-developed I topology and particularly a frequency range of the signal processing which can be higher than with circuits with the standard operational amplifiers. lassical three-port current conveyors, often denoted as II (second-generation), were introduced by Smith and Sedra []. Since their introduction (970) the II have led to a great number of applications in signal processing circuits, especially many oscillators and A filters have been given, for examples see references [] - [7]. In case of the biquadratic (secondorder) filters, mostly as basic all-pole filters, namely lowpass (L), band-pass (B), high-pass (H) type only, less commonly as notch biquads, having transmission zeros, namely as band-reject filters (B), low-pass notch (L) and high-pass notch (H) filters, what will be given in this paper. The synthesis of the conveyor based circuits is still an active topic. A unique systematic design procedure is presented, which is based on generalized autonomous network corresponding with suitable characteristic equation, as it was in detail shown in [].. Three-ort urrent onveyors It is well known that the definition relations of the II (+) port variables in Fig. are v A v, i B i, i 0. () x y z x y There ideally A B. Thus the Y port acts as high-impedance terminal and copies the voltage at the. The port is a low-impedance terminal and copies (convey the current at the port. ote that in this type II there is the absence of the current I Y () and the current I flows in the same direction of the current I (Fig. ), if it is opposite, the conveyor is of other type, namely II (-). Fig.. egative impedance converter using the II (+).. egative Impedance onverter from onveyor II (+) A unique application of the II (+) is obtained (firstly reported in [6]), if the terminals Y and are connected together (Fig.), than v v v, i i, i 0. () x y z z x y This two-port represents an impedance converter (I) with the following relations between external port variables v. () v, i i The impedance transfer is inp L ()
ADIOEGIEEIG, VOL. 6, O., SETEMBE 007 97 and the I is negative (I). hanging the input and output the same relations (), () can be obtained [9], the I is reciprocal. This I with II (+) (Fig. ) will be used down as a basic building block in our synthesis of the several biquads as follows.. Structures of Biquads Using egative Impedance onverter To design a biquadratic A filter with the I, the procedure from [] can be used. One suitable general structure (GBS) of the biquad, using single I and four passive () admittances, is shown in Fig. a. There the I separates two passive twoports and the impedance transformation improves the property of the whole biquad (increases the factor). The structure GBS- (Fig. holds the voltage transfer function in following general form YY A. () YY + Y Y + Y Y Y Y AB This formula is resulting from a routine of symbolical nodal analysis, done by the computer tool SA. The second suitable general structure (GBS-) of the biquad with five passive admittances and one I is given in Fig. b. There is a feedback by the admittance Y. The transfer function of this structure (Fig. is resulting (by the program SA) in a little complicated symbolic formula K Y Y + Y Y Y Y YY + Y Y + Y Y + Y Y Y Y Y Y. (6) ote that for simplification the ideal II (+) is assumed, so A B. The other two given structures are modifications of the GBS- only. The GBS- has the admittance Y (Fig. omitted. Then the voltage transfer function (6) has the simplified form K Y Y + Y Y Y Y. (7) YY YY The structure GBS- has also Y 0 and modified Y as a parallel connection of two admittances Y Y A + Y B. The transfer function of the general structure GBS- (Fig. c) is more complicated, the numerator for the case of A B is given by (8) and denominator by (9) ( Y Y Y + Y Y Y + Y Y Y Y Y Y, (8) D( YY Y Y Y Y +. (9) + Y Y Y + Y Y Y + Y Y Y Y Y Y Substituting a voltage buffer (VVS) in the GBS- as shown in Fig. d the last suitable general structure GBS-6 is obtained. Then the transfer function of the GBS-6 is simpler, namely the denominator is D( YY Y Y Y Y +. (0) + Y Y Y + Y Y Y YY Y and the numerator keeps the previous form (8). A suitable choosing of the admittances Y k ( or ) for realization of the several functional biquads will be shown down. ote that other modification of this general structure with single I can be possible. I Y Y I II+ Y Y Y Fig.. General biquad structures using the I: structure GBS-, structure GBS-, c) structure GBS-, d) structure GBS-6.. Low-ass All-ole Biquad with I From the general structures given above (Fig. ) we can build a frequency filters, so that we choose the admittances Y k as a single resistor k or capacitor k to obtain the desired form of the transfer function. For a low- Y c) d)
98 T. DOSTÁL, V. AMA, BIUADS BASED O SIGLE EGATIVE IMEDAE OVETE pass all-pole biquad the structure GBS- (Fig. is most suitable, choosing the concrete admittances as shown in Fig.. In this case and if A B the transfer function is G G s + s[ ( G G) + G ] + GG. () From the denominator of the () a frequency of poles (or cut-off - db frequency) () and a quality factor of poles () have been derived in following forms G G db, (). () G + G G ( G G ) These equations can be used to design the certain low-pass biquad. The optimal case is if G G. Usually, then the design equations are simpler and the quality factor is proportional to the resistor spread only,. (), () Fig.. Low-pass all-pole biquad using the I. 6. High-ass Biquad Using I The H biquad can be obtained from Fig. using the known L H transformation [7] replacing k k. The resulting circuit diagram is shown in Fig.. This circuit holds the transfer function with the same denominator s s s[ ( G G) G ], (6) + + + GG the same equations for the frequency () and the quality factor () of the poles as was given above. For the optimal design the equations (), () can also be used. I Fig.. G II+ Y High-pass biquad using the I. G Fig.. Band-pass biquad using the I: variant B-, variant B-. 7. Band-ass Biquad Using I The structure GBS- (Fig. is suitable for the B biquad, too, choosing the admittances as shown in Fig.. Symbolical analysis of the circuit B- in Fig. a, by the tool SA, is resulting in the voltage transfer function s G. (7) s + s[ ( G G) + G ] + GG The other modification B- in Fig. b was obtained interchanging k k. Then the I separates the first subcircuit L- and second one H-. The transfer function has other numerator only s G. (8) s + s[ ( G G) + G ] + GG To design the B biquads all notes given above can be used too (see section ). 8. Band-eject Biquad Using I The function of the band-reject filter (B) is to attenuate a finite band while passing both lower and higher frequencies. A second order transfer function that realizes the B has following general form K () s U U out in K 0 s + s +, (9) s + s + where to achieve a symmetric B gain response and usually (the zeros are on the imaginary axi. Then the attenuation is infinite at and controls the sharpness of the notch.
ADIOEGIEEIG, VOL. 6, O., SETEMBE 007 99 The B biquad can be realized by the structure GBS- (Fig. b with the admittance Y 0). The two possible network realizations of the B biquad are given in Fig. 6. 9. Low-ass otch Biquad Using I The low-pass notch biquad (L) can be based on the general structure GBS- (Fig., adding in the variant B- (Fig. 6 one other capacitor (Y s ), as shown in Fig. 7a. Assuming the condition G G G the general transfer function (6) is modified in this concrete form s + G. (6) s ( + ) + sg + G I G Y II+ G Fig. 6. Band-reject biquad using the I: variant B-, variant B-. The variant B- in Fig. 6a holds the following transfer function s + s( ( G G )) + G G. (0) s + s( ( G G) + G ) + GG For the desired position of the zeros, or for in the eq. (9), the condition G G G must be valid and then the transfer function (0) has the simplest form s + G. () s + sg + G omparing () and (9) the design equations are done, namely the band-reject or zero and pole frequency c) B, () and the quality factor of the poles is proportional to the capacitor spread only. () The dual variant B- (changing ) in Fig. 6b has similar properties, following the dual condition and the design equations B, (). () Fig. 7. Low-pass notch biquad based on the I: variant L-, variant L-, c) variant L-, d) variant L-. Then the variant L- (Fig. 7 holds the following parameters of the poles and zeros:, +, (7), (8) d)
00 T. DOSTÁL, V. AMA, BIUADS BASED O SIGLE EGATIVE IMEDAE OVETE +,. (9), (0) The second variant L- is based on the general structure GBS-, choosing the concrete admittances as shown in Fig. 7b. There the admittance Y is modified as a parallel connection of two admittances (Y Y A + Y B ), namely the parallel connection of and G. This circuit (Fig. 7 has the similar properties. The zeros are staying on the imaginary axis, or factor, if the desired condition + is valid. Then the transfer function is given by s + GG, () s + s G + G G and the design equations are,, (), () 0. High-ass otch Biquad with I The high-pass notch biquads (H are dual to the L s and can be obtained from Fig. 7 replacing k k. he resulting circuit diagrams are shown in Fig. 8. The variant H- is also given from the B- (Fig. 6 adding one other resistor (Y G ), as shown in Fig. 8a. Assuming the acceptable condition, the H- (Fig. 8 holds the transfer function s + GG, (6) s + sg + G ( G + G ) and the following design equations G ( G + G ), G,(7), (8) G + G G + G,. (9), (0),. (), () The other variant L- is shown in Fig. 7c. This variant is based on the general structure GBS-. There the desired requires the condition G G G and then the transfer function is K ( s G + G G. (6) s ( G + G) + sg( G + G) + G G I G G II+ Y G By this way the biquad L- (Fig. 7c) can be designed using the following, little complicated, equations +, +,(7), (8) +,. (9), (0) The last variant of the L- (Fig. 7d) is based on the general structure GBS-6, adding in the branch with the in the L- (Fig. 7c) a voltage follower (VVS, with the gain A ). In this case the transfer function (6) is modified in c) s G + G G. () s ( G + G) + sg( G + G) + G G and then the design equations are, + +, +, (), (). (), () d) Fig. 8. High-pass noth biquad based on the I: variant H-, variant H-, c) variant H-, variant H-.
ADIOEGIEEIG, VOL. 6, O., SETEMBE 007 0 To obtain the desired parameter in the variant H- (Fig. 8 it is required G G -G. Then s + GG K (, () s + s G + G and the design equations are, G, G, (), () G + G. (), () For the variant H- (Fig. 8c) the acceptable condition is and then the transfer function has the following form s + GG K (, (6) s + sg + + + GG and the design equations are +,, (7), (8) + + G,. (9), (60) G The last variant H- (Fig. 8d) is based on the general structure GBS-6 (Fig. d) with the voltage follower in one branch. Then the transfer function (6) and the design equations are modified in s + G G, (6) s, + + sg + + + G G + +, (6), (6),. (6), (6) Using these formulas we can design all types of notch biquads, namely the L-, H- and the B, too. A handicap of these simple circuits with a single II is impossibility in arbitrary value setting and some proportion between f p /f n and p.. All-ass Biquad Using I The all-pass (AF) is a special type of the biquad, whose magnitude response is constant, but the phase is linearly frequency-dependent and/or the group delay is constant in some frequency range [7]. The classical implementation of the AF biquad using standard opamp is well known, nevertheless such a network can be implemented by our I, too. The AF biquad can be obtained from the B biquad given above in Fig. 6, both variants B- or B-, taking other design conditions. The zeros in the left half-plain must be mirroring the poles. amely for the B-, the following condition must be valid G ( G G ). (66) Then the natural frequency and quality factor of the zeros and poles have the same values, given by following equations GG, G. (67), (68) G Defining the component ratios m, n, (69) then m, (70) n and the design way can be noted as, m,, n, (7) n +, n m. (7), (7) n ( ). ealization and Simulation To evaluate the performance of the structures given above, a practical band-pass filter based on the B- biquad (Fig. was designed with the commercially available components. The main functional block, which is the II (+), was realized by the input part (x, y, z) of the commercially available current feedback amplifier AD 8. The basic circuit diagram from Fig. b was ingeniously supplemented by electronic tuning, based on the bootstrap principle. The analogue multiplier MLT 0 provides a variable resistor in Fig. 9, namely grounded G (Fig., what is simpler and floating G (Fig., what needs suplementary adders using TL07. The proposed circuit in Fig. 9 was simulated with Spice using professional macro models and experimentally tested. The resulting measured magnitude responses (Fig. 0) have confirmed the symbolical analysis and theoretical assumptions.. onclusion The given structures based on the negative impedance converter implemented by the single classical three-port current conveyor II with connected ports Y- is really the right choice to design basic and notch A biquads.
0 T. DOSTÁL, V. AMA, BIUADS BASED O SIGLE EGATIVE IMEDAE OVETE. I TUIG MLT0 TL07 +I OA -I G AD8 Y II+ AD8 TL07 +I OA -I TL07 -I OA +I G GD MLT0. Fig. 9. ealization of the B biquad (Fig. ). Ku [db] 0 0-0 -0-0 -0-0 0 00 000 0000 f [Hz] Fig. 0. Magnitude responces of the the B biquad (Fig. 9). Vtun -.V Vtun 0V Vtun.V Vtun V Vtun.V These circuits need prevailingly one functional block only (the structure GBS-6 two) and maximally five passive elements. One single II means smaller chip area, lower power consumption and lower noise. The proposed biquads are expected to be used in elliptic filters at higher ranges. Acknowledgements The research described in the paper was financially supported by the zech Ministry of Education under the research program MSM 0060. eferences [] SEDA, A. S., SMITH, K.. A second generation current conveyor and its applications. IEEE Transaction on ircuit Theory, vol. T- 7, Feb. 970, p. -. [] SVOBODA, J. A., McGOY, L., WEBB, S. Applications of commercially available current conveyor. International Journal of Electronics, 99, vol. 70, no., p. 9-6. [] SU, Y., FIDLE, J. K. Versatile active biquad based on secondgeneration current conveyors. International Journal of Electronics, 99, vol. 76, p. 9-98. [] DOSTÁL, T., ČAJKA, J., VBA, K. Design procedure of oscillators and biquads based on current conveyors. In roceeding of 8 th WSEAS International onference on ircuits and Systems S 0. Athens (Greece), 00, sec. 87, paper,. [] ELMA, S., MATIE,. A., ALOSEA, A. Approach to synthesis of canonic -active oscillators using II. IEE roceeding of ircuits Devices and Systems, 99, vol., no. 6, p. 9-97. [6] FEI, G., GUEII,.. Low-Voltage Low-ower MOS urrent onveyors. Kluwer Academic ublishers, Boston, 00. [7] TOUMAOU,., LIDGEY, E. J., HAIGH, D. G. Analogue I Design: The urrent Mode Approach. eter eregrinus Ltd., London, 990. [8] HE, W. K. The ircuits and Filters Handbook. ress, Florida, 99. [9] DOSTÁL, T., AMA, V. urrent conveyor II with connected ports Y-. In roceeding of 7 th International onference adioelektronika 007. Brno (zech epublic), 007, p. 7-0. About Author... Tomáš DOSTÁL (*9) received his h.d (976) and Dr.Sc. degree (989) from the Brno University of Technology. He was with the Military Academy Brno (976-978 and 980-98) and with the Military Technical ollege Baghdad (978-980). Since 98 he has been with the Brno University of Technology as rofessor of adio-electronics. His interests are in circuit theory, analogue filters, switched capacitor networks and circuits in the current mode. Vladimír AMA (*977) received his Ing. degree (000) from the Brno University of Technology. He is with ST Microelectronics as design layout engineer. His interests are in analogue and mixed circuits and filters.