Inverse Band Reject and All Pass Filter Structure Employing CMOS CDBAs

Inverse Band Reject and All Pass Filter Structure Employing CMOS CDBAs

Ram Bhagat

Department of Electrical Engineering Delhi Technological university Delhi, India

D. R. Bhaskar

Department of Electronics and Communication Engineering Delhi Technological university

Delhi, India

Pragati Kumar

Department of Electrical Engineering Delhi Technological university Delhi, India

Abstract In this paper a new voltage-mode (VM) circuit for the realization of second order tuneable inverse band reject filter (IBRF) and inverse all pass filter (IAPF) have been introduced. The proposed circuit uses two current differencing buffered amplifiers (CDBA), two capacitors and four/five resistors. PSPICE simulation results employing CMOS CDBA implemented in 0.18µm CMOS TSMC technology have been presented to validate the workability of the configuration.

Keywords Analog Inverse Filters, Current Differencing Buffered Amplifier, Analog Signal Processing

1. INTRODUCTION

   An analog inverse filter provides frequency selective characteristics which are opposite to those of the corresponding analog filter. These inverse filters are used to remove the effect of distortion introduced in the signal while it is processed through a system. Inverse filters find applications in control, instrumentation and communication systems [1]. Recently, a number of analog inverse active filterswith different properties have been realised with different active building blocks(ABB) like operational amplifiers (op-amps) [2], current feedback operational amplifiers (CFOAs) [3]-[7], second generation current conveyors CCIIs [8], [9], four terminal floating nullors (FTFNs) [10], [11], operational transresistance amplifiers (OTRAs) [13], [14] current differencing buffered amplifiers (CDBAs) [15], [16], current differencing transconductance

   amplifiers CDTAs [17], [18], operational transconductance amplifiers OTAs [19], differential difference current conveyors (DDCCs) [20] and voltage differencing transconductance amplifiers (VDTAs) [21]. However, there are few number of research publications for realising second order IBRF [3], [4], [7], [12], [14], [16] and IAPF [11], [12]

   and [14].

   The CDBA has received prominent attention in analog signal processing and signal generation circuits due to its important property of providing at its input terminals virtual ground, which eliminates the effect of parasitic at the input terminals. Apart from this, its low output impedance enables the CDBA based circuit configurations to be cascaded easily. From existing literature survey, it has been observed that only two circuit structures have been reported so far to

   realize second-order inverse active filters employing CDBAs [15], [16]. [15] has reported two CDBA based second order inverse low pass filter (ILPF) with four resistors and two capacitrs, inverse high pass filters (IHPF) using two resistors and four capacitors and inverse band pass filter (IBPF) with three resistors and three capacitors whereas in [16] the ILPF circuit is realized with four resistors and two capacitors the other four circuits namely, IHPF, IBPF, IBRF and IAPF have been realised with 3-4 capacitors and 2-4 resistors. Also none of these two reported inverse filters have utilized the intrinsic (current differencing) property of the CDBA, as n-terminal of one of the CDBA has been left open resulting into employment of more than two capacitors for realization of second order inverse filters of different types in many cases (IBPF, IHPF, IBRF and IAPF).

   Therefore, the purpose of this communication is to present a new inverse filter circuit configuration employing two CDBAs with their intrinsic (current differencing) property fully utilized, four/five resistors and only two capacitors along with a switch. The realised second order IBRF has independent control of cut-off frequency and bandwidth while IAPF has independent tunability of cut-off frequency. The workability of the presented inverse filters has been verified using PSPICE simulations implemented in TSMC 0.18µm CMOS technology.

2. THE PROPOSED CONFIGURATION

   The symbolic representation of the CDBA is shown in Fig.1, where p and n are current input terminals (with ideally zero input impedances), w is the voltage output terminal (ideally zero output impedance) and z is the current output terminal (ideally infinite output impedance).

   The characterising terminal equations of an ideal CDBA are given by:

   Iz = (Ip – In), Vp= 0 = Vn and Vw = Vz

   (1)

   The proposed configuration for the realization of IBRF and IAPF with two capacitors and four/five resistors is shown in Fig. 2.

   	p z CDBA n w	z
   In		Iw

   	p z CDBA n w	z
   In		Iw

   Ip I

   Vp Vz

   Vn Vw

   Fig. 1 Symbolic representation of CDBA

   R1

   be tuned independently i.e. 0 through R3/C4 and BW

   through R4

3. NON-IDEAL ANALYASIS

   The non-ideal representation of the CDBA taking into account the parasitic at various terminals is shown in Fig.3, where parallel combination of Rz and 1/sCz is the parasitic impedance at port z, Rp and Rn are the parasitic resistances at terminals p and n respectively and Ro is the parasitic output resistance at terminal w.

   A practical CDBA can be described [22] by the following relationships that take into account the non-idealities of the device:

   VP Vn 0,IZ PIP nInandVW VZ

   (6)

   R5

   R5

   where P 1P and p( p 1) is the current tracking

   R3 error from p-terminal to z-terminal, n 1 n

   where

   p w

   R2 CDBA1

   n w Vo

   S CDBA2

   n( n

   1) is the current tracking error from

   1. erminal to z-

      n z p z

      terminal, and 1 v

      with

      v( v 1) is the voltage

      Vin

      C1 R4

      C2

      tracking error from -terminal to w-terminal of the CDBA. In addition to the above, if we also consider parasitic impedances at z-terminal and parasitic resistances at p, n and

      Fig. 2. The proposed inverse filer structure

      Realization of IBRF and IAPF responses from Fig. 2 is as follows:

      Case I: When switch S is open the resulting transfer function becomes

      Vo (s) 1 Vin (s) s 2 1 R1R 3C1C2 s 2 1 s 1 R 4 C2 R 2 R 3C1C6

      (2)

      which represents an IBRF with cut-off frequency (0) and

      w terminals of CDBA as shown in Fig. 3 and reanalyze the circuits of Fig. 2, we get the following non-ideal transfer functions of inverse active filters:

      The non-ideal expression of IBRF is given by

      1 1 s2 Bs C Vo(s) 2 Vin (s) s2 Ms N IBRF	(7)
      where C C C 1 z1 2 B 1 R z2 R z2 R z1 2 C1 (C2 CZ2 ) 2 CZ1 (C2 CZ2 ) C z2 R Z1 C 1 12n1P2 C (C C ) C (C C ) R R 2 1 2 Z2 2 Z1 2 Z2 1 3 2 C1 2 Cz1 R z2R z2 M 1 2 C2 2 CZ2 2 C1 (C2 CZ2 ) 2 CZ1 (C2 CZ2 ) R z1 R Z1 C4P2 Cz1P2 R R 4 4 N 1 12n1n 2 2 C (C C ) C (C C ) R R R R 2 1 2 Z2 2 Z1 2 Z2 2 3 Z2 z1

      1 1 s2 Bs C Vo(s) 2 Vin (s) s2 Ms N IBRF	(7)
      where C C C 1 z1 2 B 1 R z2 R z2 R z1 2 C1 (C2 CZ2 ) 2 CZ1 (C2 CZ2 ) C z2 R Z1 C 1 12n1P2 C (C C ) C (C C ) R R 2 1 2 Z2 2 Z1 2 Z2 1 3 2 C1 2 Cz1 R z2 R z2 M 1 2 C2 2 CZ2 2 C1 (C2 CZ2 ) 2 CZ1 (C2 CZ2 ) R z1 R Z1 C4P2 Cz1P2 R R 4 4 N 1 12n1n 2 2 C (C C ) C (C C ) R R R R 2 1 2 Z2 2 Z1 2 Z2 2 3 Z2 z1

      bandwidth (BW) given by:

      o 1 For R1 = R 2 and R 2 R 3C1C2 BW 1 R 4 C 2

      (3)

      Case II: When switch S is closed the resulting transfer function will be given by:

      Vo (s) 1 Vin (s) s2 1 s 1 R7C6 R1R3C1C2 s2 1 s 1 R4C2 R 2R3C1C2

      (4)

      Vo (s) 1 Vin (s) s2 1 s 1 R7C6 R1R3C1C2 s2 1 s 1 R4C2 R 2R3C1C2

      (4)

      o 1 For R1 = R 2 and R 2 R 3C1C2 BW 1 For R R R 4 C2 4 5

      (5)

      o 1 For R1 = R 2 and R 2 R 3C1C2 BW 1 For R R R 4 C2 4 5

      (5)

      which realizes an IAPF with 0

      and BW given by:

      The non-ideal expressions of IAPF when switch S is closed:

      Thus, it is clear that from equation (3), both 0 and BW can

      RP

      RP

      VP IP p

      RO IW

      V0(s) 1 Vin (s) 1 2 p2 12n1P2 IAPF s C R s C C R R 2 2 2 5 2 4 1 3 s2 p2 s 12n1n 2 C2R 4 C1C2R 2R3

      (11)

      V0(s) 1 Vin (s) 1 2 p2 12n1P2 IAPF s C R s C C R R 2 2 2 5 2 4 1 3 s2 p2 s 12n1n 2 C2R 4 C1C2R 2R3

      (11)

      w

      VW

      Vn In Rn

      CDBA

      n z

      IZ

      0

      0

      VZ which realizes IAPF for with , and BW given by:

      12n1n 2 0 C C 1 R R 1 2 n1 2 3 12n1n 2 For R = R O 1 1 2 n1 BW p2 BW For R = R C R P2 5 4 2 4

      (12)

      12n1n 2 0 C C 1 R R 1 2 n1 2 3 12n1n 2 For R = R O 1 1 2 n1 BW p2 BW For R = R C R P2 5 4 2 4

      (12)

      RZ CZ

      Fig. 3. Non-ideal CDBA with Parasitic elements

      1 1 s2 Bs C Vo(s) 2 Vin (s) s2 Ms N IAPF	(8)
      where C1 CZ1 C2 B 1 Rz2 Rz2 Rz1 C (C C ) C (C C ) C (C C 2 1 2 Z2 2 Z1 2 Z2 Z2 2 P2 4 Z1) R Z1 R 4 C 1 12n1P2 2P2 2C1(C2 CZ2 ) 2CZ1(C2 CZ2 ) R1R3 Rz1R4 2C4 2CZ1 2C6 M 1 Rz2 Rz2 Rz1 2C1(C2 CZ2 ) 2CZ1(C2 CZ2 ) 2CZ2 C4P2 CZ1P2 R R R Z1 4 4 N 1 12n1n2 2 2C1(C2 CZ2 ) 2CZ1(C2 CZ2 ) R2R3 RZ2Rz1

      1 1 s2 Bs C Vo(s) 2 Vin (s) s2 Ms N IAPF	(8)
      where C1 CZ1 C2 B 1 Rz2 Rz2 Rz1 C (C C ) C (C C ) C (C C 2 1 2 Z2 2 Z1 2 Z2 Z2 2 P2 4 Z1) R Z1 R 4 C 1 12n1P2 2P2 2C1(C2 CZ2 ) 2CZ1(C2 CZ2 ) R1R3 Rz1R4 2C4 2CZ1 2C6 M 1 Rz2 Rz2 Rz1 2C1(C2 CZ2 ) 2CZ1(C2 CZ2 ) 2CZ2 C4P2 CZ1P2 R R R Z1 4 4 N 1 12n1n2 2 2C1(C2 CZ2 ) 2CZ1(C2 CZ2 ) R2R3 RZ2Rz1

      Equations (10) and (12) reveal that the cut-off frequency, and bandwidth of the IBRF and IAPF are dependent on voltage/current tracking error coefficients of the CDBA terminals, hence subsequently introduce errors in the 0 and BW of the inverse filters.

4. SENSITIVITY ANALYSIS

   The active and passive sensitivities of 0 and BW of inverse active filters (IBRF and IAPF) are tabulated in Table 1 which shows that the sensitivities with respect to active and passive components are small.

   Table-1 Sensitivity analysis of inverse filters

   Filter	Sensitivity
   IBRF	SO SO 1 ,S0 SO 1 , S0 S0 0 R 2 R3 2 C1 C2 2 R1 R 4 S O S 0 S O 1 , S 0 S 0 0 1 2 n2 2 p1 p2 SBW SBW 1, S BW S BW S BW 0 R 4 C2 R2 R3 C4 S BW 0, S BW 1, S BW S BW S BW 0 p1 p2 1 2 n2
   IAPF	SO SO 1 ,S0 SO 1 , R 2 R3 2 C1 C2 2 S0 S0 SO 0 R1 R4 R5 S O S 0 S O 1 , S 0 S 0 0 1 2 n2 2 p1 p2

   Filter	Sensitivity
   IBRF	SO SO 1 ,S0 SO 1 , S0 S0 0 R 2 R3 2 C1 C2 2 R1 R 4 S O S 0 S O 1 , S 0 S 0 0 1 2 n2 2 p1 p2 SBW SBW 1, S BW S BW S BW 0 R 4 C2 R2 R3 C4 S BW 0, S BW 1, S BW S BW S BW 0 p1 p2 1 2 n2
   IAPF	SO SO 1 ,S0 SO 1 , R 2 R3 2 C1 C2 2 S0 S0 SO 0 R1 R4 R5 S O S 0 S O 1 , S 0 S 0 0 1 2 n2 2 p1 p2

   Since the value of Rz is very high and the value of Cz is very low, therefore, the inverse filter transfer functions can be approximated as:

   V0 (s) 1 Vin (s) s 2 1n1P2 IBRF R R C C 2 1 3 1 2 s 2 p2 s 1 2n1P2 C2 R 4 C1C2 R 2 R 3 ch realizes IBRF with cut-off frequency

   (9) ( 0 ) and

   V0 (s) 1 Vin (s) s 2 1n1P2 IBRF R R C C 2 1 3 1 2 s 2 p2 s 1 2n1P2 C2 R 4 C1C2 R 2 R 3 ch realizes IBRF with cut-off frequency

   (9) ( 0 ) and

   whi

   12n1n2 For R = R 0 C C R R O 1 2 n1 n 2 1 2 1 24 2 3 BW p2 BW C R p2 2 4 and

   (10)

   12n1n2 For R = R 0 C C R R O 1 2 n1 n 2 1 2 1 24 2 3 BW p2 BW C R p2 2 4 and

   (10)

   bandwidth ( BW ) given by:

5. SIMULATION RESULTS

   The validity of the proposed inverse filters has been examined through PSPICE simlations using an exemplary CMOS CDBA architecture derived from [23], shown here in Fig. 4. The DC power supplies used were ± 2.5V and the bias current was taken 40µA.The aspect ratios used for implementation of CDBA are given in Table 2. The inverse filters were designed for a nominal value of the cut-off frequency of 159 kHz and quality factor Q = 1. The passive components were selected as C1 = C2 = 100pF, R1 = R2 = R3 R4= R5 = 10k. Fig. 5 displays the frequency responses of IBRF and Fig. 6 shows the gain and phase responses of IAPF. The tunability of cut-off frequency and quality factor

   of the IBRF have been shown in Fig. 7 and Fig. 8 respectively. The input signal level during PSPICE simulations were kept at 1 VPP.

   –

   +V DD M 5

   I b

   M 6 M 1

   M 14

   I b

   3

   3

   M 7 M 8 M 15

   M 2

   M1

   M 17

   M 16

   M 23

   M 25

   p n z

   M 4

   M 3 M 18

   M 24

   w M 26

   M 19 M 20

   M 9 M 10

   I b I b

   Fig. 7. Tunability of bandwidth for different value of Q for IBRF

   – VSS

   M 11

   M 12 M 21

   M 22

   Fig.4 CMOS realization of CDBA derived from [23] TABLE -2. Aspect ratios of MOSFETs used for realization of CDBA

   MOSFETS	ratios
   M1,M2,M17,M23,M25	50/0.5
   M3,M4,M18,M24,M26	100/0.5
   M5-M16,M19-M22	3.33/0.5

   Fig. 5. Frequency responses of IBRF

   Fig. 8. Tunability of cut-off frequency for IBRF

   The Monte Carlo simulation results of the inverse active filters showing variation of gain and cut-off frequency for 5% variation in resistance and capacitance values and have been displayed in Fig. 9 and Fig. 10 respectively. For IBRF the cut-off frequency obtained from the simulations was found to be 150 kHz while Monte Carlo analysis shows that median value of cut-off frequency as 152.148 kHz and similarly for IAPF corresponding to a simulated gain of 0 dB, the Monte Carlo analysis shows that median value of gain as 0.3 dB which shows that the mismatch in the passive component values do not have large effect on the cut-off frequency and gain.

   Fig. 6. Gain and phase responses of IAPF

   P 30

   e r c e n

   t 20

   o f

   S

   a

   1. 10 p

      l e s

      20

      10

      0

      -10

      0

      120K 140K 160K 180K

      CenterFrequency(DB(V(9)),30)

      -20

      10KHz 100KHz 1.0MHz

      … DB(V(9)/V(3))

      Frequency

   2. samples = 1000 n divisions = 10

   mean = 152899

   sigma = 5927.23

   minimum = 138071

   80

   40

   0

   -40

   10th %ile = 145311 median = 152148 90th %ile = 160866 maximum = 179600

   3*sigma = 17781.7

   (a)

   (b)

   Fig. 10. Monte-Carlo analysis of IAPF showing variation of gain for 5% variation in resistance and capacitance (a) Histogram (b) frequency response.

6. CONCLUSIONS

A new VM analog circuit for realizing IBRF and IAPF response has been proposed employing two fully utilised CDBAs, two capacitors, four/five resistors and a switch. The proposed IBRF offers independent tunability of cut-off frequency and quality factor whereas, the IAPF has independent control of cut-off frequency Non-ideal analysis of both the inverse filters has revealed that there is a little error in the cut-off frequency due to inclusion of the parasitic of the CDBAs. The active and passive sensitivities of presented inverse filters are low. The validity of inverse filters has been confirmed through PSPICE simulations using an exemplary CMOS CDBA architecture implemented in TSMC 0.18µm CMOS technology. These simulation results have been found to be in good agreement with the

10KHz 100KHz 1.0MHz

… DB(V(9))

Frequency

(b)

theoretical results.

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Fig. 9. Monte-Carlo analysis of IBRF showing variation of cut-off frequency for 5% variation in resistance and capacitance (a) Histogram (b) frequency response.

P 30

e r c e n

t

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   p 10

   l

   e

   s

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   -1.0 0 1.0 2.0

   = 0.306029

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   = 0.306029

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