Limitations on input signal level in voltage-mode active-RC filters using current conveyors

Microelectronics Journal Microelectronics Journal 30 (1999) 69–76

Limitations on input signal level in voltage-mode active-RC filters using current conveyors C. Acar*, H. Kuntman Istanbul Technical University, Faculty of Electrical and Electronics Engineering, Department of Electronics and Communication Engineering, 80626, Maslak, Istanbul, Turkey

Abstract The current conveyors (CCIIs) operate linearly under certain conditions. Violation of these conditions causes nonlinear distortion in active filters involving CCIIs. On the other hand, voltage-mode active-RC filters are considerably important in filter design. Considering these facts, a simple formula is derived for maximum input signal amplitude preventing nonlinear distortion in voltage-mode active-RC filters involving current conveyors. Derived formula are verified by SPICE simulations on chosen active filter examples. Using the formula presented in this study and the impedance scaling property, the input signal level can be optimized and the behaviour of the active filters can be improved. 䉷 1998 Elsevier Science Ltd. All rights reserved. Keywords: Current conveyors; SPICE simulations; Active-RC filters

1. Introduction

2. Non-linear behaviour of the current conveyors

In realization of active filters, the designers assume that active components are linear components. In fact they are non-linear and they behave linear under certain conditions. These conditions depend on design parameters of active components. Violation of these conditions cause non-linear distortion in filters. Hence, the designers should know in advance linear operation conditions of the filter to be realized. Linear operation conditions for several active components such as OPAMPs, OTAs, CCIs and CCIIs are investigated by introducing macromodels [1–6]. But, not much study is performed as to linear operation of active filters. However, limitations on input signal level in voltage-mode OTA-C filters and CCII-based current-mode active RC filters have appeared recently in literature [7,8]. Although CCII-based voltage-mode filters are widely used in filter design, this problem is not taken into consideration in this type of filter design [9,10,12,13]. In this paper, using linear operation conditions of typical CMOS current conveyors, limitations on input signal amplitiude in voltage-mode current conveyor filters are studied. A simple formula is derived for maximum input signal amplitude not causing a non-linear operation.

The circuit representation of noninverting (CCIIþ) and inverting (CCII¹) types of second generation current conveyors are shown in Fig. 1(a) and (b). Circuit configurations [10] for CMOS realization of CCIIþ and CCII¹ are also illustrated in Fig. 1(c) and (d), respectively, where v x(t), v y(t), v z(t), i x(t), i y(t) and i z(t) are terminal voltages and currents, respectively. In ideal case CCIIs are characterized by

* Corresponding author

iy (t) ¼ 0

(1a)

vx (t) ¼ vy (t) iz (t) ¼ ⫾ ix (t) In actual case CCIIs are non-linear components and behave linearly if the following conditions are satisfied: Vxm ¹ ⱕ vx (t) ⱕ Vxm þ

(1b)

Vzm ¹ ⱕ vz (t) ⱕ Vzm þ Ixm ¹ ⱕ ix (t) ⱕ Ixm þ where Vxm þ , Vxm ¹ , Vzm þ , Vzm ¹ , I xmþ, I xm ¹ represent maximum positive and negative voltage at the x-terminal, maximum positive and negative voltage at the z-terminal and the maximum positive and negative current at the x-terminal, respectively.

0026-2692/99/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved. PII: S0 02 6 -2 6 92 ( 98 ) 00 0 88 - 3

Fig. 1. (a) Circuit represantation of an ideal CCIIþ; (b) circuit represantation of an ideal CCII¹; (c) CMOS realization of an CCIIþ; and (d) CMOS realization of an CCII¹.

70 C. Acar, H. Kuntman / Microelectronics Journal 30 (1999) 69–76

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C. Acar, H. Kuntman / Microelectronics Journal 30 (1999) 69–76

for the CMOS CCIIþ circuit in Fig. 1(c) and

Table 1 Dimensions of the MOS transistors

¹ 9:5 V ⱕ vx (t) ⱕ 7:94 V

Device

CCIIþ CCII¹ Device W(m)/L(m) W(m)/L(m)

CCIIþ CCII¹ W(m)/L(m) W(m)/L(m)

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

17/10 17/10 17/10 17/10 24/5 24/5 23/5 23/5 24/5 24/5 24/5 24/5

23/5 23/5 23/5 23/5 — — — — — — — —

17/10 17/10 17/10 17/10 24/5 24/5 23/5 23/5 24/5 24/5 24/5 24/5

M13 M14 M15 M16 M17 M18 M19 M20 M21 M22 M23 M24

23/5 23/5 23/5 23/5 24/5 24/5 23/5 23/5 24/5 24/5 23/5 23/5

for the CMOS CCII¹ circuit in Fig. 1(d), respectively.

3. Determination of the maximum input signal amplitude

(2a)

¹ 9:31 V ⱕ vz (t) ⱕ 7:17 V ¹ 204 mA ⱕ ix (t) ⱕ 1:1 mA

¹ 9:3 V ⱕ vz (t) ⱕ 9:1 V ¹ 116 mA ⱕ ix (t) ⱕ 1:112 mA

Typical transfer characteristics of noninverting and inverting current conveyors are depicted in for CMOS CCII topologies given in Fig. 1(c) and (d) [10]. The supply voltages were chosen as V DD ¼ ¹V SS ¼ 10 V. The dimensions and the model parameters of the MOS transistors in CCIIþ and CCII¹ of Fig. 1(c) and (d) are respectively given in Table 1 and Table 2. As seen from Fig. 2, there are saturations in x-terminal and z-terminal voltages and currents. If x-terminal or z-terminal voltage of any current conveyor saturates, this causes clipped waveform in filter. If x-terminal or z-terminal current of any current conveyor saturates, this causes sawtooth waveform, which is known as the slew-rate limiting problem [11]. In order to operate the current conveyors in the linear region we have to impose the following restrictions on v x, v z and i x as seen from the transfer characteristics given in Fig. 2: ¹ 9:5 V ⱕ vx (t) ⱕ 6:7 V

(2b)

For linear operations, the input signal level of the filter under consideration must be adjusted so that the above conditions be simultaneously satisfied for every current conveyor for designers specified frequency band q 僆 (q 1, q 2): Vxk ⱕ Vsxk (3a) Vzk ⱕ Vszk , k ¼ 1, 2, …, n

(3b)

Ixk ⱕ Isxk

(3c)

where n denotes the total number of current conveyors used in the design. V xk ¼ V xk(jq) and V zk ¼ V zk(jq) are respectively phasor voltages at x and z-terminal of the kth current conveyor. I xk ¼ I xk(jq) is also phasor current through x-terminal of the kth current conveyor. V sxk, V szk and I sxk are the bounds of linear region and defined as  (3d) Vsxk ¼ min Vxm þ , Vxm ¹  Vszk ¼ min Vzm þ , Vzm ¹

(3e)

 Isxk ¼ min Ixm þ , Ixm ¹

(3f)

We call these quantities saturation voltages and current of the kth current conveyor. They have values of 7.94 V, 9.1 V, 116 mA and 6.7 V, 7.17 V, 204 mA for the previous mentioned inverting (CCII¹) and noninverting (CCIIþ)

Table 2 SPICE MOS LEVEL-2 parameters for NMOS and PMOS transistors used for simulations Parameter LD VTO GAMMA UO UCRIT VMAX LAMBDA NEFF TPG CGDO CGBO MJ MJSW XQC

NMOS 0.414747U 0.864893 0.981 656 107603 100000 0.0107351 1.001 1 2.835E-10 7.968E-10 0.456300 0.3199 1

PMOS 0.580687U ¹0.944048 0.435 271 20581.4 33274.4 0.0620118 1.001 ¹1 4.831E-10 1.293E-9 0.4247 0.2185 1

Parameter

NMOS

PMOS

TOX KP PHI UEXP DELTA XJ NFS NSS RSH CGSO CJ CJSW PB NSUB

505.0 ⫻ 10 44.9 ⫻ 10 ¹6 0.6 0.211012 3.53172 0.4U 1 ⫻ 10 11 1 ⫻ 10 12 9.925 2.835 ⫻ 10 ¹10 0.0003924 5.284 ⫻ 10 ¹10 0.7 1.3563 ⫻ 10 16 ¹10

432.0 ⫻ 10 ¹10 18.5 ⫻ 10 ¹6 0.6 0.242315 4.3209 ⫻ 10 ¹5 0.4U 1 ⫻ 10 11 1 ⫻ 10 12 10.25 4.831 ⫻ 10 ¹10 0.0001307 4.613 ⫻ 10 ¹10 0.75 1 ⫻ 10 16

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Vi : Tzk ⱕ Vszk , k ¼ 1, 2, …, n Vi : Yxk ⱕ Isxk where lV il is the amplitude of the filter’s input voltage. T xk ¼ T xk(jq) is the voltage transfer function which is defined as the ratio of the kth current conveyor’s x-terminal phasor voltage to the phasor input voltage: Txk ¼

Vxk (jq) , k ¼ 1, 2, …, n Vi

(5)

T zk ¼ T zk(jq) is also the voltage transfer function defined as the ratio of the kth current conveyor’s z-terminal phasor voltage to the phasor input voltage: Tzk ¼

Vzk (jq) , k ¼ 1, 2, …, n Vi

(6)

Y xk ¼ Y xk(jq) is the transfer admittance function which is defined as the ratio of the kth current conveyor’s x-terminal phasor current to the phasor input voltage: Yxk ¼

Ixk (jq) , k ¼ 1, 2, …, n Vi

(7)

There exist 3n inequalities in Eq. (4). They put the following constraints on input voltage amplitude for q 僆 (q 1, q 2): Vi ⱕ Vsxk (8) Txk Vi ⱕ Vszk Tzk Vi ⱕ Isxk Yxk The common solution of these inequalities which gives the maximum value of the input voltage’s amplitude not causing non-linear distortion can be expressed as: V i max ( ) Vsxk Vszk Isxk ¼ min , , , k ¼ 1, 2, …, n , Txk max Tzk max Yxk max ÿ
ð9aÞ q 僆 q1 , q2

Fig. 2. Simulated DC transfer characteristics of CMOS CCIIþ and CCII¹ given in Fig. 1(c) and (d); (a) plot of V x against V y for open-circuited xterminal; (b) plot of V z against V y for open-circuited z-terminal and R X ¼ 1 k; and (c) plot of I x against V y for R X ¼ 1 k.

current conveyors, respectively. In terms of input voltage amplitude, these conditions can be written as follows: Vi : Txk ⱕ Vsxk (4)

where lT xkl max, lT zkl max and lY xkl max are respectively the maximum values of lT xkl, lT zkl and lY xkl for the designer’s specified frequency band q 僆 (q 1, q 2). In the case that the identical and the same type of current conveyors (CCIIþ or CCII¹) are used in the realization of filters, then the bounds are equal to each other and they can be denoted by V sx, V sz and I sx. In this case Eq. (9a) is converted to ( ) V V I sx sz sx Vi ¼ min , , , k ¼ 1, 2, …, n , max Txk Tzx Yxk max

ÿ

q 僆 q1 , q2




max

max

ð9bÞ

C. Acar, H. Kuntman / Microelectronics Journal 30 (1999) 69–76

73

Fig. 3. Second-order lowpass filter using three current conveyors.

Note that saturation voltages and current (i.e. V sx, V sz and I sx) are known from current conveyor topologies used in the design. Transfer voltage and transfer admittance functions (i.e. T xk, T zk and Y xk) in Eq. (9a) and Eq. (9b) are to be calculated from the given network topology of filter involving current conveyors by the use of Eq. (5), Eq. (6) and Eq. (7), respectively. In the following, the evaluation of the maximum input signal value is shown using the circuit [12] of Fig. 3. This circuit realizes a lowpass filter characteristic whose d.c. gain, pole frequency and pole quality factor are respectively T(j0) ¼ 1, q P ¼ 10 6 rad sec ¹1 and Q P ¼ 1, if the element values are chosen as:

Tx2 ¼

Vx2 G1 :G3 ¼ Vi D(jq)

Tx3 ¼

Vx3 G :(jqC2 þ G4 ) ¼¹ 1 Vi D(jq)

Tz1 ¼

Vz1 G :(jqC2 þ G4 ) ¼¹ 1 Vi D(jq)

Tz2 ¼

Vz2 G :(jqC2 þ G4 ) ¼¹ 1 D(jq) Vi

Tz3 ¼

Vz3 G1 :G3 ¼ Vi D(jq)

C1 ¼ C2 ¼ 100 pF

Yx1 ¼

Ix1 ¼ G1 Vi

The noninverting and inverting type of current conveyors in Fig. 3 are designed using the configurations [10] shown in Fig. 1. The supply voltages were chosen as V DD ¼ ¹V SS ¼ 10 V. The saturation voltages and current of

Yx2 ¼

Ix2 G1 :G2 :G3 ¼ Vi D(jq)

Vsx ¼ 6:7 V

Yx3 ¼

Ix3 G :G (jqC2 þ G4 ) ¼¹ 1 3 Vi D(jq)

R1 ¼ R2 ¼ R3 ¼ R4 ¼ 10 k

(10)

(11a)

Vsz ¼ 7:17 V

where D(jq) is characteristic polynomial of the filter:

Isx ¼ 204 mA

D(jq) ¼ (jq)2 C1 C2 þ (jq)·C1 G4 þ G2 G3

are obtained from Eq. (2a) for CCIIþ, and the saturation voltages and current of Vsx ¼ 7:94 V

(11b)

Vsz ¼ 9:1 V Isx ¼ 116 mA are obtained from Eq. (2b) for the CCII¹. The voltage transfer functions and transfer admittance functions are obtained from the filter topology in Fig. 3 as V Tx1 ¼ x1 ¼ 1 Vi

(12)

Using the previous transfer function for the passband region of q 僆 (0,q p), we obtain: Tx1 (jq) ¼ 1 (13) max

Tx2 (jq)

max

¼ 1:1516

Tx3 (jq)

max

¼ 1:464

Tz1 (jq)

max

¼ 1:464

Tz2 (jq)

max

¼ 1:464

Tz3 (jq)

max

¼ 1:1516

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C. Acar, H. Kuntman / Microelectronics Journal 30 (1999) 69–76

Yx3 (jq)

max

¼ 146:404 mA V ¹ 1

Substitution of Eq. (11a), Eq. (11b) and Eq. (13) into Eq. (9a) yields  Vi ¼ min 7:94 V, 5:82 V, 5:42 V, 6:22 V, 4:9 V, 7:9 V, max

1:16 V, 1:77 V, 0:79 Vg V i

max

¼ 0:79 V

This result is verified by the SPICE simulations of the filter. Simulated output responses of the filter to 100 kHz sinusoidal input voltages for lV il ⬍ 0.79 V, lV il ¼ 0.79 V and lV il ⬎ 0.79 V are shown in Fig. 4. Note from both the formulation presented in this paper and simulation results that the output voltage is distorted for lV il ⬎ lV il max ¼ 0.79 V. In the second example, the same analysis is repeated for a more complex filter structure, namely for the all pole bandpass filter shown in Fig. 5. This circuit realizes a third order Butterworth characteristic with unity gain at resonant frequency of q p ¼ 10 6 rad sec ¹1, if the element values are chosen as: R1 ¼ 40 kQ

R2 ¼ 20 kQ R3 ¼ 10 kQ

C1 ¼ 50 pF,

C2 ¼ 50 pF

R ¼ 20 kQ

C3 ¼ 50 pF (14)

This circuit is obtained by adding a summing amplifier to the lowpass filter topology proposed by Acar [13]. It involves only the positive type of CCIIs and they are designed using the the CMOS configuration in Fig. 1(c). Their saturation current and voltages are already given in Eq. (11a). The transfer functions which are necessary for determination of lV il max are obtained from the filter topology of Fig. 5, as Tx1 ¼

Vx1 1 ¼ Vi D(jq)

Vx2 ¼ (1 þ jqC1 b1 )·Tx1 Vi   Vx3 b2 b Tx3 ¼ ¼ 1 þ jqC3 ·Tx2 ¹ jqC3 2 Tx1 Vi b1 b1

Tx2 ¼

Tx4 ¼ Fig. 4. Simulated output response of the filter to 100 kHz sinusoidal input voltages; (a) lV il ¼ 0.5 V ⬍ lV il max; (b) lV il ¼ lV il max ¼ 0.79 V; and (c) lV il ¼ 1.25 V ⬎ lV il max.

Yx1 (jq)

max

Yx2 (jq)

¹1 max ¼ 115:162 mA V

¼ 100 mA V ¹ 1

Vx4 ¼ Tx3 Vi

Vzi ¼ 0, i ¼ 1, 2, 3 Vi p 2(jq)2 Vz4 ¼ Tz4 ¼ Vi D(jq)

Tzi ¼

Yx1 ¼

ÿ
Ix1 ¼ jqC3 Tx1 ¹ Tx2 Vi

(15)

C. Acar, H. Kuntman / Microelectronics Journal 30 (1999) 69–76

75

Fig. 5. Third-order all pole bandpass filter using four current conveyors.

Yx2 ¼

ÿ
1ÿ
Ix2 ¼ jqC2 Tx2 ¹ Tx3 þ T ¹ Tx1 Vi b1 x2
Gÿ þ p Tx2 ¹ Tx4 2

Yx3 ¼


Ix3 b1 ÿ ¼ Tx3 ¹ Tx2 V i b2

Yx4 ¼

Ix4 ¼ G:Tz4 Vi

Substitution of Eq. (11a) and Eq. (16) into Eq. (9b) yields  Vi ¼ min 6:7 V, 4 V, 4:24 V, 4:24 V, 6:98 V, 2:81 V, max

1:6 V, 2:8 V, 3:97 Vg V i

max

¼ 1:6 V

this result is also verified by the SPICE simulation of the filter.

where D(jq) is a characteristic polynomial of the filter:

4. Conclusions

D(jq) ¼ b3 (jq)3 þ b2 (jq)2 þ b1 (jq) þ 1

In this paper the maximum input signal level that does not not cause non-linear distortion as a clipping and slew-ratelimiting effect is investigated for voltage-mode active-RC filters involving second generation current conveyors. The formula of Eq. (9a) and Eq. (9b) was derived in order to determine this maximum level. Note from the given examples and this formula that the slew-rating effect is dominant in case of high frequency operations. Note also that the input signal level can be optimized and, hence, the behaviour of active-RC filters using current conveyors can be improved by using impedance scaling, since impedance scaling does not scale the voltage transfer functions T xk and T zk, but scales the transfer admittance functions Y xk for k ¼ 1; 2; …; n.

where the normalized coefficients are b 1 ¼ b 2 ¼ 2, b 3 ¼ 1. Using the previous transfer functions, for the 3 dB frequency band of q 僆 (0.65q p, 1.76q p), we obtain Tx1 (jq)

max

¼1

Tx2 (jq)

max

¼ 1:677

Tx3 (jq)

max

¼ 1:58

Tx4 (jq)

max

¼ 1:58

max

¼ 0 i ¼ 1, 2, 3

Tzi (jq)

Tz4 (jq)

max

¼ 1:027

Yx1 (jq)

max

¼ 72:516 mA V ¹ 1

Yx2 (jq)

max

¼ 127:636 mA V ¹ 1

Yx3 (jq)

max

¼ 72:658 mA V ¹ 1

Yx4 (jq)

max

¼ 51:334 mA V ¹ 1

(16)

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