Int. J. Electron. Commun. (AEÜ) 107 (2019) 192–198
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Short communication
Novel CFOA based capacitance multiplier and its application Rakesh Verma, Neeta Pandey ⇑, Rajeshwari Pandey Department of Electronics and Communication Engineering, Delhi Technological University, Bawana Road, Delhi 110042, India
a r t i c l e
i n f o
Article history: Received 12 January 2019 Accepted 8 May 2019
Keywords: CFOA Capacitance multiplier Resonance Multiplication factor
a b s t r a c t This communication presents a novel CFOA based capacitance multiplier. The proposed circuit employs two CFOAs, two resistors and a single capacitor. To achieve higher multiplication factors, the larger component spread is needed in existing CFOA based capacitance multiplier circuits. The proposal addresses this and attains multiplication at lower component spread. The mathematical analysis for modelling the effects of non-ideality shows the deviations from ideal behavior which may be compensated by placing an additional resistor. The functionality is tested using SPICE simulations and experimentation under different conditions. An application namely parallel RLC resonator is included to show the usefulness. Ó 2019 Elsevier GmbH. All rights reserved.
1. Introduction Large valued capacitors are imperative for implementing zeromaking capacitor in loop filter in monolithic phase locked loop (PLL) [1] and applications pertaining to low frequency signal processing such as sample and holds, switched-capacitor, continuous-time filters [2] and implantable biomedical systems [3]. The on-chip realization of such capacitors is not practical [1,2] due to significant area overhead. In order to realize large capacitors, the concept of capacitance multiplier may be used where active blocks are arranged in a configuration providing a multiplication factor for a small on-chip capacitance [3]. In the last few decades a variety of grounded/floating capacitance multiplier circuits [4–43] have been presented using active blocks like operational amplifiers (Op-amps) [4,5], current follower transconductance amplifiers (CFTAs) [6], voltage differencing buffered amplifiers (VDBAs) [7–9], current backward transconductance amplifiers (CBTAs) [10], operational transconductance amplifiers (OTAs) [11–13], OTAs with op-amp [14], current feedback operational amplifiers (CFOAs) [15–20], tunable four terminal floating nullors (TFTFNs) [21] and second-generation current conveyors (CCIIs)/current-controlled CCIIs [22–35]/generalization of current conveyors (CCs) namely dual-X CCII (DXCCII) [36], CC transconductance amplifier (CCTA) [37], differential voltage CCTAs (DVCCTAs) [38,39], differential voltage CCs (DVCCs) [27,40,41], current controlled CC transconductance amplifier (CCCCTAs) [42], current controlled differential difference CCs (CCDDCCs) [43], voltage differencing CCs (VDCCs) [44], fully differential (FD) voltage ⇑ Corresponding author. E-mail address:
[email protected] (N. Pandey). https://doi.org/10.1016/j.aeue.2019.05.010 1434-8411/Ó 2019 Elsevier GmbH. All rights reserved.
and current gained (VCG)-CCIIs [45], and current differencing transconductance amplifier (CDTA) [46]. It may be noted that though numerous capacitance multipliers exist but very few are based on commercially available active blocks namely Op-amp [4,5,14], OTA [11–14], AD 844 (CFOA) [15–20,32] and CDTA [46]. This contribution aims at proposing a CFOA based capacitance multiplier circuit. The paper is organized as follows: A brief review of available CFOA multipliers is placed in Section 2. The proposed capacitance multiplier circuit is put forward in Section 3. To examine operation of the proposed circuit, SPICE simulations are carried out under different test cases and the corresponding results are included in Section 3. This section also includes experimental results. An application of proposed multiplier circuit namely reconfiguring parallel RLC resonance circuit is placed in Section 4. The functionality of the application is verified through SPICE simulation which clearly demonstrates the scaling behavior of reconfigured resonator on the basis of numerous case studies. Section 5 is the conclusion part. 2. Brief literature review on CFOA based capacitance multipliers The study of CFOA based capacitance multiplier (C-multiplier) [15–20,32] shows that the multiplication factor (K) of capacitance (C) can be expressed in the form of (i) 1 + P [16,32], (ii) 1-P [32] and (iii) 1/(1 + P) [16,32] where P represents resistor ratio. The structures of type (ii) may be used to realize a C-multiplier presenting a negative capacitance value if P is greater than unity and a positive value for P less than unity. The capacitance multiplier circuit may be obtained by adaptation of CFOA based gyrator [47–49]/generalized impedance converter [50]. Other characteristics are summarized below:
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– [16,18] provide lossy capacitance whereas lossless is obtained in [15,17–20,32] – [15,18,20,32] realize grounded capacitance while floating C realization are found in [16,17,19] – [16,32] can emulate both positive and negative C-multiplier while positive C occurs in [15,32] and negative C in [17–20,32] – The multiplication factor of C-multipliers of type (iii) and type (ii) with P > 1 is less than unity. So, the effective value of capacitor decreases which may be used multiplication factor multiplication factor multiplication factor capacitor which is not available otherwise. – Larger component spread (resistance ratio) is needed to achieve higher multiplication factor [15–20,32,47–50]. It is clear from above discussion that available CFOA based topologies require larger component spread for higher multiplication factor. An alternative approach is presented in this paper wherein multiplication factor depends on ratio of resistance and difference between the resistor values. Therefore, closer are the resistor values, higher will be the multiplication factor.
3. Proposed CFOA based capacitance multiplier circuit This section presents a new type of C-multiplier circuit with multiplication factor of K = 1/(1 P) and designate this as type (iv). This type of circuit can provide very high multiplication factor by selecting P close to unity. The closer is P to unity, higher would be the multiplication factor resulting in smaller component spread. The CFOA symbol and port relationships are described first and followed by proposed circuit schematic and related discussion. 3.1. The CFOA
3.2. Proposed circuit The schematic of proposed C-multiplier circuit is shown in Fig. 2. It uses two CFOAs, two resistors and a capacitor (for noncompensating circuit). The proposed circuit uses floating capacitor which may be realized using Metal-insulator-metal (MIM) or metal-oxide-metal (MOM) double poly (poly1-poly2) capacitor processes [51]. Considering port relation of (1), the input impedance of the proposed circuit is computed as
Z in ðsÞ ¼
1 1 1 R2 =R1 ¼ ¼ sC eff sðKC Þ sC
ð3Þ
where K ¼ 1=ð1 R2 =R1 Þ It is clear from (3) that the smaller is the component spread larger will be the multiplication factor (K) e. g. R2 = 0.9 R1 gives K = 10. R2 =R1 The sensitivity of the (K) with respect to R2 =R1 is SKR2 =R1 ¼ 1R . 2 =R1
Therefore, the advantage comes at the cost of higher sensitivity of K. To analyze the proposed circuit’s behavior in presence of CFOA non-idealities shown in Fig. 1(b), the input impedance is recomputed as
Z in ðsÞjn ¼
1 sC þ GY1 þ sC Y1 þ ðG
ð4Þ where GY1 ; C Y1 ; GZ1 ; C Z1 ; RX1 ; and GZ2 ; C Z2 ; RX2 are parasitics of CFOA1; and CFOA2 respectively. Considering RX2 GZ1 1 operating frequency x< min RY1 ðC1 þCÞ ; RZ21C ; RX21C , (3) reduces to Z2
Y1
Z1
1 ða2 bcÞsC
The symbol of CFOA shown in Fig. 1(a) has port relationship given by (1).
Z in ðsÞjn
IY ¼ 0; V X ¼ V Y ; IZ ¼ IX ; V O ¼ V Z
which gives multiplication factor as
ð1Þ
Fig. 1(b) shows equivalent circuit with CFOA non-idealities. The (RY, CY) and (RZ, CZ) correspond to parasitic resistor and capacitor at Y and Z terminals while RX represent parasitic resistor at X terminal. There are two voltage buffers between Y and X-terminals; and O and Z-terminals; and one current follower between Z and X-terminals where a represents current transfer gain; and b, c correspond to voltage transfer gains due to tracking errors of CFOA. This restructured port relationship of (1) as
IY ¼ 0; V X ¼ bV Y ; IZ ¼ aIX ; V O ¼ cV Z
ð2Þ
ða2 bcÞsC
2 2 þGZ2 þsC Z2 ÞðR1 þRX1 Þð1þRX2 GZ1 þsRX2 C Z1 Þa c
sC þ ðG
ð5Þ
2 2 þGZ2 ÞðR1 þRX1 Þa c
Kn ¼ 1 þ
a2 bc
ð6Þ
ðG2 þ GZ2 ÞðR1 þ RX1 Þ a2 c
It may be observed from (6) that multiplication factor (K n Þ drastically decreases for a; b; c < 1. To compensate this, a grounded 0
resistor R at X-terminal of CFOA1 may be placed (as shown in Fig. 2) which modifies (4) to
1
0
Z in ðsÞjn ¼
sC þ GY1 þ sC Y1 þ
Ideally these values are GY1 ¼ GZ1 ¼ GZ2 ¼ RX1 ¼ RX2 ¼ C Y1 ¼ C Z1 ¼ C Z2 ¼ 0 and a ¼ b ¼ c ¼ 1
Fig. 1. (a) CFOA symbol and (b) its equivalent circuit with parasitic.
ða2 bcÞsC
RX1 ðG2 þGZ2 þsC Z2 Þð1þRX2 GZ1 þsRX2 C Z1 Þa2 c
0 R 0 R1 þR
ð7Þ
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The relation between R1 and R’ is obtained by comparing (8) and (9) as 0
R ¼
P R1 1P
ð10Þ
where
RX1 ðG2 þ GZ2 Þð1 þ RX2 GZ1 Þ R1 P¼b þ 1 a2 bc R2
ð11Þ
The behavior of capacitance multiplier circuit is examined under non-compensating, compensating and ideal conditions and simulation results are demonstrated in Fig. 3. The responses are plotted for factor K = 100 where R1 = 20.02 kO, R2 = 20 kO, 0
R = 32.16 O, by assuming non-ideal characteristics occurred due to a; b; c = 0.99 and RX1 ¼ RX2 = 50 O, RY1 ¼ RY2 = 2 MO, RZ1 ¼ RZ2 = 3 MO, C Y1 ¼ C Y2 = 2 pF, C Z1 ¼ C Z2 = 4.5 pF. Fig. 2. Proposed CFOA based capacitance multiplier circuit.
Considering R G 1, operating frequency x< X2 Z1 min RZ21C ; RX21C ; and neglecting parasitic effect at Y-terminal, Z2
Z1
(7) reduces to
1 ða2 bcÞsC
0
Z in ðsÞjn
sC þ
RX1 ðG2 þGZ2 Þð1þRX2 GZ1 Þa2 c
0
ð8Þ
R 0 R1 þR
3.3. Functional verification The functionality of the proposed capacitance multiplier is examined using CFOA model [52] using the following simulations conditions: (i) K is fixed and varying C; (ii) C is fixed and varying K; and (iii) K and C are fixed and varying temperature.
Since ideally it is expressed by (9)
Z in ðsÞ ¼
1 sC þ G2 RsC1 1
ð9Þ
which are designated as case 1, case 2 and case 3 respectively; and detailed simulation settings are placed in Table 1. Simulated and theoretical frequency responses are plotted in Fig. 4(a) and
Fig. 3. Magnitude responses of proposed non-compensating and compensating circuits with ideal case.
Table 1 Detailed simulation settings and summary of observations. Case
Components tuning C (nF)
1
2
0.1 10 100 1
3
1
Realized Ceff (F)
K tuning
Temperature (°C)
K
R1 (kO)
R2 (kO)
R0 (kO)
100
20.202
20
4.4
27
50 100 200 5
20.408 20.202 20.1 25
20 20 20 20
9.1 4.4 2.23 (not required)
27
55 27 125
10n 1m 10m 50n 100n 200n 5n
Frequency response Magnitude response within 7% error
Phase response within 60 phase error
upto 390 kHz upto 43.6 kHz upto 25.1 kHz upto 141 kHz upto 144.5 kHz upto 190 kHz 5.75 Hz–512.8 kHz upto 281 kHz upto 331 kHz
upto 67.6 kHz upto 20.5 kHz upto 2.89 kHz upto 75.8 kHz upto 55 kHz upto 40.7 kHz 10 Hz–794 kHz upto 316.23 kHz upto 416.8 kHz
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Fig. 4. Simulation (solid lines) and ideal (dashed lines) magnitude (in black) and phase (in green) responses of the proposed Fig. 2 circuit for (a) fixed K = 100 and different C = 100 nF, 10 nF, 0.1 nF (b) fixed C = 1 nF, and different K = 50, 100, 200. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(b) respectively for cases 1 and 2 at room temperature. The effective value of capacitance and frequency range is also listed in Table 1. Fig. 5 shows the simulation and theoretical frequency responses by varying temperature.
The functionality of the proposed circuit is examined through experimentation. The proposed circuit is breadboarded for K = 5 and 21 with (R1 = 25 kX, R2 = 20 kX) and (R1 = 21 kX, R2 = 20 kX) respectively. Power supply of ±10 V is considered and a sinusoid
Fig. 5. Impedance magnitude and phase responses of the proposed Fig. 2 circuit at different temperature.
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Fig. 6. Experimental results of Fig. 2 for K = 5 and 21.
having Vpeak-peak of 2 V is applied to examine the performance of the proposal. The simulated, theoretical and experimental magnitude response is depicted in Fig. 6 for C = 1 nF. The frequency ranges of multiplier circuit for K = 5 and 21 are (0.02–2.8) kHz and (9.1–10.9) kHz respectively within 10% deviations. A deviation of up to 40% is observed for K = 21 for frequencies up to 20 kHz. The designer needs to take care of limited range for higher capacitance value i.e. designer has to preadjust the value of resistor once the frequency/frequency range of operation is fixed. It is emphasized here that high multiplication factor requires passive components with great precision. Therefore it is difficult to achieve very high multiplication factor. In integrated circuit realization, such precision may be achieved by using electronically tunable resistor.
The impedance function of the reconfigured parallel RLC circuit is given as
Z in ¼
1 1 ¼ Y in R1 þ sL1 þ sC eff eff
ð13Þ
eff
and its performance parameters resonant frequency (x0 ), quality-factor (Q) and peak impedance at resonant frequency (Z in jx0 ) are evaluated as
x0 ¼
qffiffiffiffiffiffiffiffiffiffi 1 Leff C eff
Q ¼ Reff :
qffiffiffiffiffiffi
ð14Þ
C eff Leff
Z in jx0 ¼ Reff 0
Assuming K1 and K2 as multiplication factors for C eff and C eff i.e.
4. Application
0
The proposed C multiplier circuit (Fig. 2) may be add extra degree of freedom for scaling the circuit behavior. In this section, the proposed circuit is employed for reconfiguration of a parallel RLC resonator (Fig. 7) for simulating grounded parallel RL combination using active CFOA element [53]. The values of components Reff and Leff are given in (12) while C eff is obtained from (2) 0
Reff ¼
0
R1 R2
0
0
0
R1 þ R2
0
0
; Leff ¼ C eff R1 R2
ð12Þ
0
C eff ¼ K 1 C, and C eff ¼ K 2 C , the performance parameters of (14) can be rearranged as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 0 0 0 : K 2 C R1 R2 K 1 C pffiffiffiffiffiffiffi ffi 0 0 qffiffiffiffiffiffiffiffi R R Q ¼ 0 1 0 2 : K 1 C0
x0 ¼
R1 þR2
Z in jx0 ¼
K2 C
0 0 R1 R2 0 0 R1 þR2
Fig. 7. (a) Reconfiguration of parallel RLC resonance circuit (b) CFOA based active simulation of parallel RL circuit.
ð15Þ
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R. Verma et al. / Int. J. Electron. Commun. (AEÜ) 107 (2019) 192–198 Table 2 Performance case study of the reconfigured parallel RLC resonator. Case
Case Test no.
Reff
C eff
0
1
2
3
S1 S2 S3 S4 S5 S6 S7 S8 S9
0
Leff 0
R1 ¼ R2 (kO)
Reff (kO)
K1
C (nF)
C eff (nF)
K2
C (nF)
Leff (H)
0.5 1 2 0.5
0.25 0.5 1 0.25
100
1
100
100
1
1
0.25
200 100 50 100 200 400
200 100 50 200 100 50
1
0.5
200 100 50 50 100 200
0.025 0.1 0.4 0.05 0.025 0.0125 0.1 0.05 0.025
2
2
x0 (krad/s)
Q
Z in jx0 (kO)
20 10 5 10 20 40 10
0.5
0.25 0.5 1 0.25
0.5
0.25 0.5 1
0.25
Fig. 8. Simulation outputs for input impedance of reconfigured parallel RLC (Fig. 7) for (a) Case1, (b) Case 2, (c) Case 3.
The performance parameters of the reconfigured resonator can be tuned by controlling the effective values of components (C eff , Leff , Reff ) of the reconfigured resonator. Following test cases are selected: 0
0
Case 1: Tuning R1 and R2 while keeping K 1 and K 2 constant Case 2: Tuning K 1 and K 2 while maintaining K 1 ¼ K 2 Case 3: Tuning K 1 and K 2 while retaining their product (K 1 :K 2 ) constant Case 1 affects the performance parameters x0 and Z in jx0 and is thus useful when these parameters need be changed for the fixed value of Q-factor. Case 2 is useful for changing x0 while retaining the Q-factor and Z in jx0 . Case 3 is appropriate for varying Q-factor and keeping x0 and Z in jx0 constant. Simulations for all the three cases are carried out with test conditions listed in Table 2 and the input impedance under the test conditions are plotted in Fig. 8. The performance parameters are also listed in Table 2. Thus its performance parameters can be independently controlled by following observations obtained from above simulations:
– Case1: different desirable impedance peaks (=0.25 k, 0.5 k, 1 k) and fixed Q (=0.5) at different x0 (=20 krad/s, 10 krad/s, 5 krad/s) – Case2: x0 tuning (=10 krad/s, 20 krad/s, 40 krad/s) with constant impedance peaks (=0.25 k) and Q (=0.5) – Case3: Q tuning (=0.25, 0.5, 1) with constant impedance peaks (=0.25 k) and x0 (=10 krad/s). 5. Conclusion A CFOA based capacitance multiplier is presented in this paper which uses two CFOAs, two resistors and a single capacitor. It requires lower component spread in order to achieve larger multiplication factor. Effects of non-idealities are addressed and compensation method is also suggested. The operation of proposed circuit is verified through SPICE simulations under variety of test cases and usefulness is illustrated through parallel RLC resonator. Declaration of Competing Interest The authors declared that there is no conflict of interest.
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