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Tunable Ladder-Type Realization of Current-Mode Elliptic Filters
International Journal of Electronics and Communications
© Urban & Fischer Verlag http://www.urbanfischer.de/journals/aeue
Tunable Ladder-Type Realization of Current-Mode Elliptic Filters Serdar Özo˘guz and Ali Toker Abstract It is shown that a new series FDNR-R equivalent circuit using current controlled current conveyors (CCCIIs) allows effective realization of jω-axis zeros which are known to complicate the active implementation of ladder-type elliptic filters. The resulting ladder-type elliptic filter employs all grounded capacitors while keeping the number of active elements small. Also, the filter parameters are electronically tunable, which is important from integration point of view. Keywords Current controlled current conveyors, Ladder filters, Elliptic filters
1. Introduction As a current-mode active device, the second-generation current conveyor (CCII) offers a number of advantages, e.g. greater linearity, wider bandwidth and design flexibility over conventional voltage-mode active devices, like opamps [1, 2]. As a result of this, many CCII-based active filters have been proposed in the literature [2]. On the other hand, a new active element, current controlled current conveyor (CCCII) has been recently introduced [3], allowing the design of filters with electronically controllable filter parameters, while offering all the advantages of the conventional CCII. Considering also the absence of the external resistors in these filters, the CCCII-based filters seem to be good options to the OTA/transconductance-C filters for the realizations of IC filters. On the other hand, the excellent filtering properties of the elliptic functions as well as the advantages of the doubly terminated LC ladder filters, i.e. very low pass-band sensitivities and low component spread, are well known [4]. Hence, various OTA–C and transconductance-C high-order elliptic ladder filters are proposed in the literature [5–8]. Some of the proposed elliptic ladder filters employ larger number of active elements compared to, for example, all-pole ladder filter realizations, due to the difficulties in realizing the transmissions zeros of the elliptic functions at the jω−axis [5, 6]. Also, any attempts for reducing the excessive number of active elements lead to the use of floating capacitors [7, 8].
Received May 25, 2001. S. Özo˘guz and A. Toker, Istanbul Technical University, Faculty of Electrical and Electronics Engineering, Department of Electronics and Communication Eng., 80626, Maslak, Istanbul, Turkey. Phone: +90-212-2853552, Fax: +90-212-2853679. E-Mail: [email protected]; [email protected] Correspondence to Ali Toker. ¨ 56 (2002) No. 3, 193−199 Int. J. Electron. Commun. (AEU)
The aim of this work is to propose a series FDNR-R equivalent circuit using CCCIIs suitable for the realization of ladder type elliptic filters. The resulting filter not only employs all grounded capacitors and small number of active elements, but also offers all the main advantages of the OTA/transconductance-C filters such as electronic control of the important filter parameters and absence of the external resistors. Moreover, the inherent dynamic range limitations of OTA/transconductance elements are achieved by using CCCIIs as active elements. Finally, the presented current-mode ladder filter is expected to offer all the potential advantages of currentmode circuits such as inherent wider bandwidth, simpler circuitry, lower power consumption and wider dynamic range [1, 9] over the known elliptic ladder filters operating in voltage-mode [5–8]. These features make the proposed filter more suitable for IC technology compared to these already known elliptic filter realizations. The effects of the CCCII active nonidealities on the filter performance are also studied and it is shown that the main limitation arises from the finite bandwidth of the CCCII current gains. The compensation methods for reducing these effects are also proposed.
2. CCCII-based subcircuits for the simulation of the prototype filter In a recent paper, Fabre et.al. introduced a new active element, CCCII [3]. The related circuit realizations and its equivalent circuit are shown in Fig. 1 and the constitutive relations of the CCCII are given as
Iy Vx Iz
0 0 0 = 1 Rx 0 0 ±1 0
Vy Ix Vz
(1)
where Rx = VT /2I0 . The positive sign in the third row denotes the positive type controlled conveyor, CCCII+ and the negative sign denotes the negative type controlled conveyor, CCCII−. From (1), it is seen that the internal resistance Rx is adjustable through the biasing current I0 . As reported in [10], the relation between the conductance of this resistor and the biasing current remains linear over two or three decades of the current, I0 . In Fig. 2(a), the proposed series FDNR-R circuit is shown. Routine analysis yields the driving-point im1434-8411/02/56/3-193 $15.00/0
194 Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters
(b)
(a) Fig. 1. a) The circuit realizations for the CCCII+ and CCCII−, b) Equivalent circuit of CCCII.
pedance function of this circuit as: Z in,1 = RxR +
s2 R
1 xD C D1 C D2
(2)
where Rx is the internal x-terminal resistance of the CCCIIs, which are controllable through the biasing current I0 . Since the impedance function in (2) has zeros at jω-axis, the circuit in Fig. 2(a) can be used to realize transmission zeros of the ladder-type elliptic ladder filters. Also, as the series equivalent resistance and the value of FDNR can be controlled by I0R and I0D , the place of the jω-axis zeros of the resulting elliptic filter can be accurately realized. From Fig. 2(a), it is seen that the zand y-terminal parasitic capacitances of CCCIIs are completely accommodated by the externally connected capacitors C D1 and C D2 . The effects of all the other CCCII nonidealities on the filter performance are studied in details in Section 5. Also, in order to simulate the series branches of the ladder-type filter, an active subcircuit realizing a float-
ing resistor is required. For this purpose, the circuit in Fig. 2(b), which simulates a floating equivalent resistance, whose value is equal to intrinsic resistance of the CCCII is considered. In the followings, we will show how an electronically controllable active-C circuit can be systematically obtained by simulating a doubly terminated LC laddertype elliptic filter using these subcircuits.
3. Elliptic filter realization using CCCIIs Consider the fifth-order doubly terminated passive LCladder filter shown in Fig. 3(a). By applying first the source transformation [11] and second the Bruton-transformation [12] to this circuit, one may obtain the prototype circuit shown in Fig. 3(b). By simulating the series and parallel branches involving resistors with the subcircuits of Fig. 2, an active-C circuit realizing this prototype circuit can readily be obtained. While simulat-
(a)
(b) Fig. 2. Subcircuits suitable for the simulation of the ladder-type elliptic LC filter.
Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters 195
alized by simulating the corresponding ladder-type prototype circuits. -
4. Design of the tuning circuitry for the proposed elliptic ladder filter
(a)
(b) Fig. 3. a) The LC elliptic ladder filter. b) The prototype circuit.
ing the FDNR-R element in the parallel branches using the subcircuit of Fig. 2(a), the transmissions zeros at the jω-axis are realized in a very convenient way, that is without having to use any additional active elements or any network using floating capacitors. It should be noted that this fact greatly simplifies the implementation of the filter. In this way, this active-C filter uses a very small number of CCCIIs and a minimum number of capacitors. On the other hand, in order to have a circuit using only grounded capacitors, the current of CL , i.e. Iout should be retrieved from a high-impedance output. This task can easily be achieved by taking the replica of the z-terminal current of the CCCII, which simulates the floating resistor R5 . The active-C circuit thus obtained, which realizes a fifth-order elliptic filter function is depicted in Fig. 4. Obviously, using the above-described approach, elliptic functions of arbitrary orders can be re-
-
-
From Fig. 4, it is seen that each CCCII has a different internal resistance, implying that each CCCII has a different biasing current. However, this does not prevent the feasibility of a proper tuning circuit. Since the value of the internal resistance, Rx is linearly controllable through the biasing current, I0 , the CCCIIs can be biased through a single controlling current, Ictrl . A possible tuning circuit for biasing CCCIIs with different internal resistances is shown in Fig. 5. The arrows indicate the inputs of the current mirrors and the current gain of the current mirror with single-input and three-outputs is unity. Also the current gains of the single-input single-output mirrors should be taken as k j = R A /R j
j = 1, 2, 3
(3)
where R j and R A are the internal resistances of the CCCIIs biased with the currents I0R j and Ictrl , respectively. In this way, the values of the internal resistances of all the CCCIIs can be adjusting through a single controlling current, Ictrl . The inaccuracies in the current gains of the singleinput single-output current mirrors may cause some deviations in the nominal values of the passive components in the ladder filter. However, it should be noted that, these deviations are not expected to have a serious effect on the filter characteristics, thanks to the very low passband sensitivities of the ladder filters. Nevertheless, as will be explained in section 6, these current mirrors with gains different than unity can be implemented using transistors with equal dimensions, reducing the in-
-
-
Fig. 4. Current-mode ladder type elliptic active-C circuit.
for
196 Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters accuracies of these current mirrors due to transistor mismatches.
5. The effects of the CCCII active nonidealities In this section, the effects of the active nonidealities of the CCCIIs on the filter performance are studied in details and some compensation methods are proposed. The main nonidealities of the CCCII can be modelled by the following hybrid equation Iy 0 0 0 Vy Vx = β(s) Rx 0 Ix (4) Iz 0 ±α(s) 0 Vz
1 1 1 − 2s/ωi 2 (1 + s/ωi ) (1 + 2s/ωi )
where β(s) =
α0 β0 , α(s) = 1 + s/ων 1 + s/ωi
that for decreasing values of I0 , the transconductances of the transistors at the CCCII output current mirrors decrease. As a result of this, especially, due to limited high-frequency performances of the lateral PNP transistors in the AMS 0.8 µ BiCMOS process, the parasitic poles proportional to the transconductances of these transistors appear at smaller frequencies. Obviously, the use of the vertical PNP transistors would relax these frequency limitations and allow the implementation of CCCII-based filters operating over hundred of MHz [10]. Anyway, for relatively smaller values of I0 , the frequency dependency of the current gain, α(s) degrades the filter performance for the process we have used and these effects are studied in the followings. Assuming CCCIIs to be equal and using the following approximations for the sake of simplicity,
(5)
Here, β0 and α0 represent the DC gains errors and their nominal values are unity. Also, ων and ωi are 3 dB cut-off frequencies of the transfer functions modelling the finite bandwidths of the voltage and current gains of the CCCII, respectively. The circuits in Fig. 1(a) are simulated in SPICE using AMS 0.8 µ BiCMOS (BSIM level 7) model parameters. β0 and α0 are found to be 0.999 and 0.985 for the CCCII+ and 0.999 and 1.009 for the CCCII−. Also, 3 dB cutoff frequency for β(s), f ν , is found to be approximately 2 GHz for both types of CCCIIs. After several simulations, it is concluded that the effects of the DC gain errors and the finite bandwidths of β(s) on the filter performance are very small thanks to the excellent sensitivity property of the ladder filter. On the other hand, it is noted that 3 dB cut-off frequency of α(s), f i , depends strongly on the value of the biasing current, I0 , which should be adjusted in order to have various values of Rx . As an example, for the CCCII−, f i is found to be approximately as 50 Mhz for I0 = 50 µA and only 10 Mhz for I0 = 10 µA (corresponding to Rx ≈ 1 kΩ). Therefore, for some small values of I0 , the bandwidth of α(s) gets smaller, which may be problematic for some applications. The reason for this is
Fig. 5. Tuning circuit for biasing CCCIIs with different Rx s from a single controlling current.
reanalysis of the subcircuit in Fig. 2(a), which simulates a series FDNR-R circuit, yields the followings drivingpoint impedance function, neglecting the effects of β(s)’s: α2 (s) s2 RxD C D1 C D2 1 2 1 ≈ RxR + 2 − s RxD C D1 C D2 sRxD C D1 C D2 ωi
Z in,1 = RxR +
(6)
From Eqs. (2) and (6), it is seen that an extra capacitor with a negative value appears in series with the simulated series resistor and FDNR, due to the frequency dependencies of the current gains of the CCCIIs. After applying the inverse Bruton transformation to the simulated circuit of Fig. 3(b), it is seen that these negative valued capacitors turn out to be as negative valued resistors in series with the inductors in the parallel branches of the circuit in Fig. 3(a). On the other hand, one may verify that, due to the finite bandwidth of α(s) of the CCCIIs included in the subcircuit of Fig. 2(b), resistors with negative values appear in parallel with the inductors in the series branches of the circuit in Fig. 3(a). All these facts mean that inductors in the parallel and series branches of the doubly terminated lossless LC ladder network in Fig. 3(a), have negative Q-factors for real CCCIIs. The effects of the inductors with the negative Q-factors on the ladder-type filter performance are well known and are also discussed in [5]. At this point, it should be noted that the effects of
Fig. 6. The biasing circuit for operating CCCIIs in Class A.
Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters 197
is not significant due to the excellent sensitivity properties of ladder-type filters. Also, the latter is completely eliminated by the negative Q-factors of these equivalent elements due to the CCCII finite bandwidths.
-
-
6. Simulation results
Fig. 7. The subcircuit simulating the parallel branches with the compensation resistor.
the CCCII finite bandwidths on the filter characteristic are similar to those of the finite bandwidths of transconductance elements in transconductance-C filters [5]. In the followings, two compensation methods, which can be used to reduce these effects, are described. First, the output current mirrors are operated in Class A in order to reduce the dependencies of the transistor transconductances at the output current mirrors to the biasing current, I0 . For this purpose, the circuit shown in Fig. 6 is connected between nodes A and B shown in Fig. 1. In this way, biasing current of the output current mirrors becomes Ibias + I0 , so that 3 dB cut-off frequency of the current gain becomes larger than a certain value, which is mainly defined by the current, Ibias . Second, in Fig. 2(a), a compensation resistor is added in series with the capacitor C D1 . As a result of this, an additional positive valued capacitor appears in series with the equivalent series FDNR-R circuit of Fig. 2(a). Using Eq. (6) and routine analysis, it is easy to show that the negative capacitance occurred due to the finite bandwidth of the current gains is completely compensated if the following value is assigned to this additional compensation resistor: Rcomp =
2 C D1 ωi
The fifth-order elliptic lowpass filter shown in Fig. 4 is modified as explained above and is simulated using SPICE AMS 0.8 µ BiCMOS (BSIM level 7) model parameters. In the simulations, we have used the CCCIIs in Fig. 1, which are supplied under ±5 V DC. As explained above, the CCCIIs are modified using the subcircuit shown in Fig. 6. Since Ibias is taken as 50 µA, the bandwidth of the current gain becomes approximately 50 MHz. The capacitor values are chosen as C D21 = C D22 = 90 pF, C D41 = C D42 = 55 pF, C S = C L = 100 pF. The CCCIIs in Fig. 4 are biased so that the internal x-terminal resistances have the following values: R1 = 1562 Ω, R2 = R D2 = 836 Ω, R3 = 1554 Ω, R4 = 3320 Ω, R5 = R D4 = 886 Ω. The corresponding tuning circuit is designed using the approach illustrated in Fig. 5. In order to reduce the effects of the transistor mismatching, the current mirrors with non-unity gains are realized using equal sized transistors. A possible way to achieve this task is to approximate the gain of the current mirror given by Eq. (3) by a ratio of two integers, say, p/q. If this current mirror is a simple one, then the diode-connected transistor of this current mirror is implemented from q parallel connected transistors and the output transistor is implemented from p transistors connected in parallels. In this way, all the current mirrors with gains different than unity are implemented from unit sized transistors. If, R A is taken as R4 , the gains of these current mirrors become: k1 = R4 /R1 = 17/8, k2 = R4 /R2 = R4 /R D2 = 4, k3 = R4 /R3 = 15/7, k4 = R4 /R4 = 1, k5 = R4 /R5 = R4 /R D4 = 15/4. Also, the value of the controlling cur-
(7)
It should be noted that a slight overcompensation may be useful for the compensation of the negative-Q factors of the serial branches inductors. This compensation resistor can be realized using the subcircuit shown in Fig. 2(b). The circuit obtained after this modification is redrawn in Fig. 7, which illustrates the implementation of this additional compensation resistor. Note that thanks to this scheme, by the use of a suitable Q-control tuning loop [13], the errors due to finite bandwidth can be corrected. It should be noted that the CCCII parasitic nonidealities, i.e. y- and z-terminal parasitic resistances and capacitances are not included in the model of Eq. (4). Due to these parasitics, the values of the equivalent capacitors and inductors in the circuit of Fig. 3(a) are deviated from their nominal values and their quality factors become finite and positive. However, note that the former
Fig. 8. Simulation results of the current-mode elliptic filter in Fig. 4. —- Ideal —– Simulation
198 Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters rent, Ictrl is calculated from Eq. (1) and is found to be approximately as 3.9 µA. The current mirrors are realized from PMOS transistors whose model parameters are extracted from the same BiCMOS process. The aspect ratios of these unit sized transistors are taken as W/L = 2 µ/4 µ. On the other hand, as explained above, the finite bandwidths of the CCCIIs involved in the subcircuits realizing the parallel branches are compensated using two additional compensation resistors connected in series with the capacitors C D21 and C D41 . From Eq. (7), the ratio of these resistors is found to be equal to the ratio of the capacitors C D21 and C D41 . Obviously, this ratio also determines the ratio of the biasing currents of the CCCIIs, which are used to realize these compensation resistors. The biasing current of the CCCII realizing the compensation resistor connected in series with the capacitor C D41 , I0,comp , is taken as 70 µA. The biasing current of the other CCCII corresponding to the resistor in series with the capacitor C D21 was C D21 /C D41 times greater than I0,comp . In this way, both parallel branches are compensated through a single controlling current, I0,comp , whose value can be properly defined by the use of a suitable Q-control tuning loop [13]. The double-output CCCII− in Fig. 4 is realized from the CCCII− in Fig. 1 by replacing the improved Wilson current mirror pairs at its z-terminal with single-input double-output cascode current mirrors. The fifth-order elliptic filter function designed as explained above has a cut-off frequency of 1.59 MHz and a passband ripple of 1 dB. In the simulations, the value of the controlling current, Ictrl is taken as 3.7 µA in order to compensate the slight deviation of the cut-off frequency due to the several parasitics related to CCCIIs. The simulation results shown in Fig. 8 verify the theoretical predictions.
7. Conclusion A current-mode elliptic ladder-type filter realization using CCCIIs is described. The filter offers the following advantages: i) Suitability for IC technology because of the absence of external resistors and electronic controllability of the filter parameters ii) The use of smaller number of active elements compared to the conventional OTA/transconductance-C elliptic filter using all grounded capacitors [5, 6] iii) The use of all grounded capacitors contrary to the some existing OTA/transconductance-C elliptic filters, which employ the same number of active elements [7, 8]. To be specific, the circuits presented in [6, 8] are given for simulating third-order elliptic functions. For this purpose, the circuits proposed in [5, 6] employ 7 active elements with all grounded capacitors, whereas those given in [7, 8] employs 4 and 5 active elements, but both uses a floating capacitor. However, for the third-order filter realization, the circuit proposed in this
work employs only 4 active elements with all grounded capacitors. Also, the effects of the bandwidth limitations, which mainly stem from the use of the lateral PNP transistors in the employed process, are discussed in details and a simple compensation method is proposed. The simulation results verifying theoretical analysis are also provided. In brief, for the implementations of elliptic ladder filters, CCCIIs seem to be good options to OTAs and transconductance amplifiers as they combine some very good properties of these active elements with the design flexibility of current conveyors. This fact allows the realization of a very low passband sensitivity elliptic active-C filter suitable for IC technology, which employs a reduced number of active elements and all grounded capacitors.
References [1] Roberts, G. W.; Sedra, A. S.: All current-mode frequency selective circuits. Electron. Lett. 25 (1989), 759–761. [2] Wilson, B.: Recent developments in current conveyor and current-mode circuits. Proc. IEE, Part-G 137 (2) (1990), 63–77. [3] Fabre, A.; Saaid, O.; Wiest, F.; Boucheron, C.: High frequency applications based on a new current controlled conveyor. IEEE Trans. Circuits and Systems-I 43 (1996), 82–91. [4] Orchard, H. J.: Inductorless filters. Electron. Lett. 2 (1966), 224–225. [5] Tan, M. A.; Schaumann, R.: Simulating general-parameter LC-ladder filters for monolithic realizations with only transconductance elements and grounded capacitors. IEEE Trans. Circuits and Systems 36 (1989), 299–307. [6] Hwang, Y. S.; Liu, S. I.; Wu, D. S.; Wu, Y. P.: Table-based linear transformation filters using OTA–C techniques. Electron. Lett. 30 (1994), 2021–2022. [7] Tan, M. A.; Schaumann, R.: A reduction in the number of active components used in transconductance grounded capacitor filters. Proc. of IEEE Int. Symp. Circuits and Systems, (1990). 2276–2278. [8] Hwang, Y. S.; Chiu, W.; Liu, S. I.; Wu, D. S.; Wu, Y. P.: High-frequency linear transformation elliptic filters employing minimum number of OTAs. Electron. Lett. 31 (1995), 1562–1564. [9] Toumazou, C.; Lidgey, F. J.; Haigh, D. G.: Analog IC design: the current-mode approach. U.K.: Peter Peregrinus on behalf of IEE, 1990. [10] Fabre, A.; Saaid, O.; Wiest, F.; Boucheron, C.: High frequency high-Q BiCMOS current-mode bandpass filter and mobile communication application. IEEE J. Solid-State Circuits SC-33 (1998), 614–625. [11] Schaumann, R.; Ghausi, M. S.; Laker, K. R.: Design of analog filters: passive, active RC and switched capacitor. New Jersey, U.S.A.: Prentice Hall Series in Electrical and Computer engineerin, 1990. [12] Bruton, L. T.: Network transfer function using the concepts of frequency-dependent negative resistance. IEEE Trans. Circuit Theory 16 (1969), 406–408. [13] Chiou, C. F.; Schaumann, R.: Design and performance of a fully integrated bipolar 10.7 Mhz analog bandpass filter. IEEE Trans. Circuits and Systems 33 (1986), 116–124.
Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters 199 Serdar Özo˘guz was born in Istanbul, Turkey, 1968. He received the B.Sc. degree in electrical & communication enginerering from the Faculty of Electrical and Electronics Eng., Istanbul Technical University, Turkey in 1991. He received the M.Sc. and Ph.D. degrees in 1993 and 2000, respectively, from the Institute of Science and Technology of the same university. He is currently an assistant professor in electronics. He is also the author or co-author of about 50 papers published in scientific reviews or conference proceedings. His main research interest is the design of analog signal processing circuits.
Ali Toker was born in Istanbul, Turkey, 1951. He received the B.Sc. and M.Sc. degrees in electrical engineering from the Faculty of Electrical and Electronics Eng., Istanbul Technical University, Turkey in 1973 and 1975, respectively. He received the Ph.D. degree in 1986 from the Institute of Science and Technology of the same university. He is currently an associate professor in electronics, teaching graduate and undergraduate courses. He is also the author or co-author of about 55 papers published in scientific reviews or conference proceedings. His main research interests are design of current-mode circuits and analog signal processing applications.
© Urban & Fischer Verlag http://www.urbanfischer.de/journals/aeue
Tunable Ladder-Type Realization of Current-Mode Elliptic Filters Serdar Özo˘guz and Ali Toker Abstract It is shown that a new series FDNR-R equivalent circuit using current controlled current conveyors (CCCIIs) allows effective realization of jω-axis zeros which are known to complicate the active implementation of ladder-type elliptic filters. The resulting ladder-type elliptic filter employs all grounded capacitors while keeping the number of active elements small. Also, the filter parameters are electronically tunable, which is important from integration point of view. Keywords Current controlled current conveyors, Ladder filters, Elliptic filters
1. Introduction As a current-mode active device, the second-generation current conveyor (CCII) offers a number of advantages, e.g. greater linearity, wider bandwidth and design flexibility over conventional voltage-mode active devices, like opamps [1, 2]. As a result of this, many CCII-based active filters have been proposed in the literature [2]. On the other hand, a new active element, current controlled current conveyor (CCCII) has been recently introduced [3], allowing the design of filters with electronically controllable filter parameters, while offering all the advantages of the conventional CCII. Considering also the absence of the external resistors in these filters, the CCCII-based filters seem to be good options to the OTA/transconductance-C filters for the realizations of IC filters. On the other hand, the excellent filtering properties of the elliptic functions as well as the advantages of the doubly terminated LC ladder filters, i.e. very low pass-band sensitivities and low component spread, are well known [4]. Hence, various OTA–C and transconductance-C high-order elliptic ladder filters are proposed in the literature [5–8]. Some of the proposed elliptic ladder filters employ larger number of active elements compared to, for example, all-pole ladder filter realizations, due to the difficulties in realizing the transmissions zeros of the elliptic functions at the jω−axis [5, 6]. Also, any attempts for reducing the excessive number of active elements lead to the use of floating capacitors [7, 8].
Received May 25, 2001. S. Özo˘guz and A. Toker, Istanbul Technical University, Faculty of Electrical and Electronics Engineering, Department of Electronics and Communication Eng., 80626, Maslak, Istanbul, Turkey. Phone: +90-212-2853552, Fax: +90-212-2853679. E-Mail: [email protected]; [email protected] Correspondence to Ali Toker. ¨ 56 (2002) No. 3, 193−199 Int. J. Electron. Commun. (AEU)
The aim of this work is to propose a series FDNR-R equivalent circuit using CCCIIs suitable for the realization of ladder type elliptic filters. The resulting filter not only employs all grounded capacitors and small number of active elements, but also offers all the main advantages of the OTA/transconductance-C filters such as electronic control of the important filter parameters and absence of the external resistors. Moreover, the inherent dynamic range limitations of OTA/transconductance elements are achieved by using CCCIIs as active elements. Finally, the presented current-mode ladder filter is expected to offer all the potential advantages of currentmode circuits such as inherent wider bandwidth, simpler circuitry, lower power consumption and wider dynamic range [1, 9] over the known elliptic ladder filters operating in voltage-mode [5–8]. These features make the proposed filter more suitable for IC technology compared to these already known elliptic filter realizations. The effects of the CCCII active nonidealities on the filter performance are also studied and it is shown that the main limitation arises from the finite bandwidth of the CCCII current gains. The compensation methods for reducing these effects are also proposed.
2. CCCII-based subcircuits for the simulation of the prototype filter In a recent paper, Fabre et.al. introduced a new active element, CCCII [3]. The related circuit realizations and its equivalent circuit are shown in Fig. 1 and the constitutive relations of the CCCII are given as
Iy Vx Iz
0 0 0 = 1 Rx 0 0 ±1 0
Vy Ix Vz
(1)
where Rx = VT /2I0 . The positive sign in the third row denotes the positive type controlled conveyor, CCCII+ and the negative sign denotes the negative type controlled conveyor, CCCII−. From (1), it is seen that the internal resistance Rx is adjustable through the biasing current I0 . As reported in [10], the relation between the conductance of this resistor and the biasing current remains linear over two or three decades of the current, I0 . In Fig. 2(a), the proposed series FDNR-R circuit is shown. Routine analysis yields the driving-point im1434-8411/02/56/3-193 $15.00/0
194 Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters
(b)
(a) Fig. 1. a) The circuit realizations for the CCCII+ and CCCII−, b) Equivalent circuit of CCCII.
pedance function of this circuit as: Z in,1 = RxR +
s2 R
1 xD C D1 C D2
(2)
where Rx is the internal x-terminal resistance of the CCCIIs, which are controllable through the biasing current I0 . Since the impedance function in (2) has zeros at jω-axis, the circuit in Fig. 2(a) can be used to realize transmission zeros of the ladder-type elliptic ladder filters. Also, as the series equivalent resistance and the value of FDNR can be controlled by I0R and I0D , the place of the jω-axis zeros of the resulting elliptic filter can be accurately realized. From Fig. 2(a), it is seen that the zand y-terminal parasitic capacitances of CCCIIs are completely accommodated by the externally connected capacitors C D1 and C D2 . The effects of all the other CCCII nonidealities on the filter performance are studied in details in Section 5. Also, in order to simulate the series branches of the ladder-type filter, an active subcircuit realizing a float-
ing resistor is required. For this purpose, the circuit in Fig. 2(b), which simulates a floating equivalent resistance, whose value is equal to intrinsic resistance of the CCCII is considered. In the followings, we will show how an electronically controllable active-C circuit can be systematically obtained by simulating a doubly terminated LC laddertype elliptic filter using these subcircuits.
3. Elliptic filter realization using CCCIIs Consider the fifth-order doubly terminated passive LCladder filter shown in Fig. 3(a). By applying first the source transformation [11] and second the Bruton-transformation [12] to this circuit, one may obtain the prototype circuit shown in Fig. 3(b). By simulating the series and parallel branches involving resistors with the subcircuits of Fig. 2, an active-C circuit realizing this prototype circuit can readily be obtained. While simulat-
(a)
(b) Fig. 2. Subcircuits suitable for the simulation of the ladder-type elliptic LC filter.
Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters 195
alized by simulating the corresponding ladder-type prototype circuits. -
4. Design of the tuning circuitry for the proposed elliptic ladder filter
(a)
(b) Fig. 3. a) The LC elliptic ladder filter. b) The prototype circuit.
ing the FDNR-R element in the parallel branches using the subcircuit of Fig. 2(a), the transmissions zeros at the jω-axis are realized in a very convenient way, that is without having to use any additional active elements or any network using floating capacitors. It should be noted that this fact greatly simplifies the implementation of the filter. In this way, this active-C filter uses a very small number of CCCIIs and a minimum number of capacitors. On the other hand, in order to have a circuit using only grounded capacitors, the current of CL , i.e. Iout should be retrieved from a high-impedance output. This task can easily be achieved by taking the replica of the z-terminal current of the CCCII, which simulates the floating resistor R5 . The active-C circuit thus obtained, which realizes a fifth-order elliptic filter function is depicted in Fig. 4. Obviously, using the above-described approach, elliptic functions of arbitrary orders can be re-
-
-
From Fig. 4, it is seen that each CCCII has a different internal resistance, implying that each CCCII has a different biasing current. However, this does not prevent the feasibility of a proper tuning circuit. Since the value of the internal resistance, Rx is linearly controllable through the biasing current, I0 , the CCCIIs can be biased through a single controlling current, Ictrl . A possible tuning circuit for biasing CCCIIs with different internal resistances is shown in Fig. 5. The arrows indicate the inputs of the current mirrors and the current gain of the current mirror with single-input and three-outputs is unity. Also the current gains of the single-input single-output mirrors should be taken as k j = R A /R j
j = 1, 2, 3
(3)
where R j and R A are the internal resistances of the CCCIIs biased with the currents I0R j and Ictrl , respectively. In this way, the values of the internal resistances of all the CCCIIs can be adjusting through a single controlling current, Ictrl . The inaccuracies in the current gains of the singleinput single-output current mirrors may cause some deviations in the nominal values of the passive components in the ladder filter. However, it should be noted that, these deviations are not expected to have a serious effect on the filter characteristics, thanks to the very low passband sensitivities of the ladder filters. Nevertheless, as will be explained in section 6, these current mirrors with gains different than unity can be implemented using transistors with equal dimensions, reducing the in-
-
-
Fig. 4. Current-mode ladder type elliptic active-C circuit.
for
196 Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters accuracies of these current mirrors due to transistor mismatches.
5. The effects of the CCCII active nonidealities In this section, the effects of the active nonidealities of the CCCIIs on the filter performance are studied in details and some compensation methods are proposed. The main nonidealities of the CCCII can be modelled by the following hybrid equation Iy 0 0 0 Vy Vx = β(s) Rx 0 Ix (4) Iz 0 ±α(s) 0 Vz
1 1 1 − 2s/ωi 2 (1 + s/ωi ) (1 + 2s/ωi )
where β(s) =
α0 β0 , α(s) = 1 + s/ων 1 + s/ωi
that for decreasing values of I0 , the transconductances of the transistors at the CCCII output current mirrors decrease. As a result of this, especially, due to limited high-frequency performances of the lateral PNP transistors in the AMS 0.8 µ BiCMOS process, the parasitic poles proportional to the transconductances of these transistors appear at smaller frequencies. Obviously, the use of the vertical PNP transistors would relax these frequency limitations and allow the implementation of CCCII-based filters operating over hundred of MHz [10]. Anyway, for relatively smaller values of I0 , the frequency dependency of the current gain, α(s) degrades the filter performance for the process we have used and these effects are studied in the followings. Assuming CCCIIs to be equal and using the following approximations for the sake of simplicity,
(5)
Here, β0 and α0 represent the DC gains errors and their nominal values are unity. Also, ων and ωi are 3 dB cut-off frequencies of the transfer functions modelling the finite bandwidths of the voltage and current gains of the CCCII, respectively. The circuits in Fig. 1(a) are simulated in SPICE using AMS 0.8 µ BiCMOS (BSIM level 7) model parameters. β0 and α0 are found to be 0.999 and 0.985 for the CCCII+ and 0.999 and 1.009 for the CCCII−. Also, 3 dB cutoff frequency for β(s), f ν , is found to be approximately 2 GHz for both types of CCCIIs. After several simulations, it is concluded that the effects of the DC gain errors and the finite bandwidths of β(s) on the filter performance are very small thanks to the excellent sensitivity property of the ladder filter. On the other hand, it is noted that 3 dB cut-off frequency of α(s), f i , depends strongly on the value of the biasing current, I0 , which should be adjusted in order to have various values of Rx . As an example, for the CCCII−, f i is found to be approximately as 50 Mhz for I0 = 50 µA and only 10 Mhz for I0 = 10 µA (corresponding to Rx ≈ 1 kΩ). Therefore, for some small values of I0 , the bandwidth of α(s) gets smaller, which may be problematic for some applications. The reason for this is
Fig. 5. Tuning circuit for biasing CCCIIs with different Rx s from a single controlling current.
reanalysis of the subcircuit in Fig. 2(a), which simulates a series FDNR-R circuit, yields the followings drivingpoint impedance function, neglecting the effects of β(s)’s: α2 (s) s2 RxD C D1 C D2 1 2 1 ≈ RxR + 2 − s RxD C D1 C D2 sRxD C D1 C D2 ωi
Z in,1 = RxR +
(6)
From Eqs. (2) and (6), it is seen that an extra capacitor with a negative value appears in series with the simulated series resistor and FDNR, due to the frequency dependencies of the current gains of the CCCIIs. After applying the inverse Bruton transformation to the simulated circuit of Fig. 3(b), it is seen that these negative valued capacitors turn out to be as negative valued resistors in series with the inductors in the parallel branches of the circuit in Fig. 3(a). On the other hand, one may verify that, due to the finite bandwidth of α(s) of the CCCIIs included in the subcircuit of Fig. 2(b), resistors with negative values appear in parallel with the inductors in the series branches of the circuit in Fig. 3(a). All these facts mean that inductors in the parallel and series branches of the doubly terminated lossless LC ladder network in Fig. 3(a), have negative Q-factors for real CCCIIs. The effects of the inductors with the negative Q-factors on the ladder-type filter performance are well known and are also discussed in [5]. At this point, it should be noted that the effects of
Fig. 6. The biasing circuit for operating CCCIIs in Class A.
Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters 197
is not significant due to the excellent sensitivity properties of ladder-type filters. Also, the latter is completely eliminated by the negative Q-factors of these equivalent elements due to the CCCII finite bandwidths.
-
-
6. Simulation results
Fig. 7. The subcircuit simulating the parallel branches with the compensation resistor.
the CCCII finite bandwidths on the filter characteristic are similar to those of the finite bandwidths of transconductance elements in transconductance-C filters [5]. In the followings, two compensation methods, which can be used to reduce these effects, are described. First, the output current mirrors are operated in Class A in order to reduce the dependencies of the transistor transconductances at the output current mirrors to the biasing current, I0 . For this purpose, the circuit shown in Fig. 6 is connected between nodes A and B shown in Fig. 1. In this way, biasing current of the output current mirrors becomes Ibias + I0 , so that 3 dB cut-off frequency of the current gain becomes larger than a certain value, which is mainly defined by the current, Ibias . Second, in Fig. 2(a), a compensation resistor is added in series with the capacitor C D1 . As a result of this, an additional positive valued capacitor appears in series with the equivalent series FDNR-R circuit of Fig. 2(a). Using Eq. (6) and routine analysis, it is easy to show that the negative capacitance occurred due to the finite bandwidth of the current gains is completely compensated if the following value is assigned to this additional compensation resistor: Rcomp =
2 C D1 ωi
The fifth-order elliptic lowpass filter shown in Fig. 4 is modified as explained above and is simulated using SPICE AMS 0.8 µ BiCMOS (BSIM level 7) model parameters. In the simulations, we have used the CCCIIs in Fig. 1, which are supplied under ±5 V DC. As explained above, the CCCIIs are modified using the subcircuit shown in Fig. 6. Since Ibias is taken as 50 µA, the bandwidth of the current gain becomes approximately 50 MHz. The capacitor values are chosen as C D21 = C D22 = 90 pF, C D41 = C D42 = 55 pF, C S = C L = 100 pF. The CCCIIs in Fig. 4 are biased so that the internal x-terminal resistances have the following values: R1 = 1562 Ω, R2 = R D2 = 836 Ω, R3 = 1554 Ω, R4 = 3320 Ω, R5 = R D4 = 886 Ω. The corresponding tuning circuit is designed using the approach illustrated in Fig. 5. In order to reduce the effects of the transistor mismatching, the current mirrors with non-unity gains are realized using equal sized transistors. A possible way to achieve this task is to approximate the gain of the current mirror given by Eq. (3) by a ratio of two integers, say, p/q. If this current mirror is a simple one, then the diode-connected transistor of this current mirror is implemented from q parallel connected transistors and the output transistor is implemented from p transistors connected in parallels. In this way, all the current mirrors with gains different than unity are implemented from unit sized transistors. If, R A is taken as R4 , the gains of these current mirrors become: k1 = R4 /R1 = 17/8, k2 = R4 /R2 = R4 /R D2 = 4, k3 = R4 /R3 = 15/7, k4 = R4 /R4 = 1, k5 = R4 /R5 = R4 /R D4 = 15/4. Also, the value of the controlling cur-
(7)
It should be noted that a slight overcompensation may be useful for the compensation of the negative-Q factors of the serial branches inductors. This compensation resistor can be realized using the subcircuit shown in Fig. 2(b). The circuit obtained after this modification is redrawn in Fig. 7, which illustrates the implementation of this additional compensation resistor. Note that thanks to this scheme, by the use of a suitable Q-control tuning loop [13], the errors due to finite bandwidth can be corrected. It should be noted that the CCCII parasitic nonidealities, i.e. y- and z-terminal parasitic resistances and capacitances are not included in the model of Eq. (4). Due to these parasitics, the values of the equivalent capacitors and inductors in the circuit of Fig. 3(a) are deviated from their nominal values and their quality factors become finite and positive. However, note that the former
Fig. 8. Simulation results of the current-mode elliptic filter in Fig. 4. —- Ideal —– Simulation
198 Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters rent, Ictrl is calculated from Eq. (1) and is found to be approximately as 3.9 µA. The current mirrors are realized from PMOS transistors whose model parameters are extracted from the same BiCMOS process. The aspect ratios of these unit sized transistors are taken as W/L = 2 µ/4 µ. On the other hand, as explained above, the finite bandwidths of the CCCIIs involved in the subcircuits realizing the parallel branches are compensated using two additional compensation resistors connected in series with the capacitors C D21 and C D41 . From Eq. (7), the ratio of these resistors is found to be equal to the ratio of the capacitors C D21 and C D41 . Obviously, this ratio also determines the ratio of the biasing currents of the CCCIIs, which are used to realize these compensation resistors. The biasing current of the CCCII realizing the compensation resistor connected in series with the capacitor C D41 , I0,comp , is taken as 70 µA. The biasing current of the other CCCII corresponding to the resistor in series with the capacitor C D21 was C D21 /C D41 times greater than I0,comp . In this way, both parallel branches are compensated through a single controlling current, I0,comp , whose value can be properly defined by the use of a suitable Q-control tuning loop [13]. The double-output CCCII− in Fig. 4 is realized from the CCCII− in Fig. 1 by replacing the improved Wilson current mirror pairs at its z-terminal with single-input double-output cascode current mirrors. The fifth-order elliptic filter function designed as explained above has a cut-off frequency of 1.59 MHz and a passband ripple of 1 dB. In the simulations, the value of the controlling current, Ictrl is taken as 3.7 µA in order to compensate the slight deviation of the cut-off frequency due to the several parasitics related to CCCIIs. The simulation results shown in Fig. 8 verify the theoretical predictions.
7. Conclusion A current-mode elliptic ladder-type filter realization using CCCIIs is described. The filter offers the following advantages: i) Suitability for IC technology because of the absence of external resistors and electronic controllability of the filter parameters ii) The use of smaller number of active elements compared to the conventional OTA/transconductance-C elliptic filter using all grounded capacitors [5, 6] iii) The use of all grounded capacitors contrary to the some existing OTA/transconductance-C elliptic filters, which employ the same number of active elements [7, 8]. To be specific, the circuits presented in [6, 8] are given for simulating third-order elliptic functions. For this purpose, the circuits proposed in [5, 6] employ 7 active elements with all grounded capacitors, whereas those given in [7, 8] employs 4 and 5 active elements, but both uses a floating capacitor. However, for the third-order filter realization, the circuit proposed in this
work employs only 4 active elements with all grounded capacitors. Also, the effects of the bandwidth limitations, which mainly stem from the use of the lateral PNP transistors in the employed process, are discussed in details and a simple compensation method is proposed. The simulation results verifying theoretical analysis are also provided. In brief, for the implementations of elliptic ladder filters, CCCIIs seem to be good options to OTAs and transconductance amplifiers as they combine some very good properties of these active elements with the design flexibility of current conveyors. This fact allows the realization of a very low passband sensitivity elliptic active-C filter suitable for IC technology, which employs a reduced number of active elements and all grounded capacitors.
References [1] Roberts, G. W.; Sedra, A. S.: All current-mode frequency selective circuits. Electron. Lett. 25 (1989), 759–761. [2] Wilson, B.: Recent developments in current conveyor and current-mode circuits. Proc. IEE, Part-G 137 (2) (1990), 63–77. [3] Fabre, A.; Saaid, O.; Wiest, F.; Boucheron, C.: High frequency applications based on a new current controlled conveyor. IEEE Trans. Circuits and Systems-I 43 (1996), 82–91. [4] Orchard, H. J.: Inductorless filters. Electron. Lett. 2 (1966), 224–225. [5] Tan, M. A.; Schaumann, R.: Simulating general-parameter LC-ladder filters for monolithic realizations with only transconductance elements and grounded capacitors. IEEE Trans. Circuits and Systems 36 (1989), 299–307. [6] Hwang, Y. S.; Liu, S. I.; Wu, D. S.; Wu, Y. P.: Table-based linear transformation filters using OTA–C techniques. Electron. Lett. 30 (1994), 2021–2022. [7] Tan, M. A.; Schaumann, R.: A reduction in the number of active components used in transconductance grounded capacitor filters. Proc. of IEEE Int. Symp. Circuits and Systems, (1990). 2276–2278. [8] Hwang, Y. S.; Chiu, W.; Liu, S. I.; Wu, D. S.; Wu, Y. P.: High-frequency linear transformation elliptic filters employing minimum number of OTAs. Electron. Lett. 31 (1995), 1562–1564. [9] Toumazou, C.; Lidgey, F. J.; Haigh, D. G.: Analog IC design: the current-mode approach. U.K.: Peter Peregrinus on behalf of IEE, 1990. [10] Fabre, A.; Saaid, O.; Wiest, F.; Boucheron, C.: High frequency high-Q BiCMOS current-mode bandpass filter and mobile communication application. IEEE J. Solid-State Circuits SC-33 (1998), 614–625. [11] Schaumann, R.; Ghausi, M. S.; Laker, K. R.: Design of analog filters: passive, active RC and switched capacitor. New Jersey, U.S.A.: Prentice Hall Series in Electrical and Computer engineerin, 1990. [12] Bruton, L. T.: Network transfer function using the concepts of frequency-dependent negative resistance. IEEE Trans. Circuit Theory 16 (1969), 406–408. [13] Chiou, C. F.; Schaumann, R.: Design and performance of a fully integrated bipolar 10.7 Mhz analog bandpass filter. IEEE Trans. Circuits and Systems 33 (1986), 116–124.
Serdar Özo˘guz, Ali Toker: Tunable Ladder-Type Realization of Current-Mode Elliptic Filters 199 Serdar Özo˘guz was born in Istanbul, Turkey, 1968. He received the B.Sc. degree in electrical & communication enginerering from the Faculty of Electrical and Electronics Eng., Istanbul Technical University, Turkey in 1991. He received the M.Sc. and Ph.D. degrees in 1993 and 2000, respectively, from the Institute of Science and Technology of the same university. He is currently an assistant professor in electronics. He is also the author or co-author of about 50 papers published in scientific reviews or conference proceedings. His main research interest is the design of analog signal processing circuits.
Ali Toker was born in Istanbul, Turkey, 1951. He received the B.Sc. and M.Sc. degrees in electrical engineering from the Faculty of Electrical and Electronics Eng., Istanbul Technical University, Turkey in 1973 and 1975, respectively. He received the Ph.D. degree in 1986 from the Institute of Science and Technology of the same university. He is currently an associate professor in electronics, teaching graduate and undergraduate courses. He is also the author or co-author of about 55 papers published in scientific reviews or conference proceedings. His main research interests are design of current-mode circuits and analog signal processing applications.