# Voltage/Current-Mode Multifunction Filters Using Current-Feedback Amplifiers and Grounded Capacitors

Authors: Jiun Wei Horng; Pei-Young Chou; Jian-Yu Wu
Journal: Active and Passive Electronic Components
 (2010)

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## Abstract

One configuration for realizing voltage-mode multifunction filters and another
configuration for realizing current-mode multifunction filters each using two current-feedback amplifiers (CFAs) are presented. The proposed voltage-mode circuit exhibits simultaneously lowpass, bandpass, and highpass filters. The proposed
current-mode circuit also exhibits simultaneously lowpass, bandpass, and highpass filters. The proposed circuits offer the following features: no requirements for component matching conditions, low active and passive sensitivities, employing only
grounded capacitors, and the ability to obtain multi-function filters from the same circuit configuration.


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## Body

## 1. Introduction

The applications and advantages in the realization of active filter transfer functions using current-feedback amplifiers (CFAs) have received considerable attention. This amplifier can provide constant bandwidth (independent of closed-loop gain) and a high slew-rate capability. Thus, it is beneficial to use a CFA as a basic building block to realize various analogue signal-processing circuits.In 1992 [1], Fabre proposed a voltage-mode bandpass and highpass filters circuit by using two CFAs, one grounded capacitor, one floating capacitor, and three resistors. In 1993 [2], Fabre proposed another voltage-mode or current-mode biquads. The voltage-mode biquad exhibits simultaneously bandpass and highpass filters by using one CFA, one grounded capacitor, one floating capacitor, and two resistors. The current-mode biquad exhibits simultaneously bandpass and highpass filters by using one CFA, two grounded capacitors, and two resistors. In 1995 [3], Liu proposed four voltage-mode biquads with high input impedance for realization lowpass, bandpass or highpass filters by using two CFAs, two (or three) capacitors, and three (or two) resistors. However, only one filter function can be obtained in each realization. Moreover, two topologies of Liu’s circuits used floating capacitors.In 1996 [4], Soliman proposed many voltage-mode biquadratic filter circuits. The four two-CFAs biquads in [4] realize lowpass and bandpass filters simultaneously and using only grounded capacitors. The two three-CFAs biquads in [4] realize lowpass, bandpass, and highpass filters simultaneously and using only grounded capacitors. In 1997 [5], Horng and Lee proposed a voltage-mode biquad for realizing lowpass, bandpass, and highpass filters simultaneously by using three CFAs, three grounded capacitors and four resistors. In 1998 [6], Senani proposed another voltage-mode biquad for realizing lowpass, bandpass and highpass filters simultaneously by using three CFAs, two grounded capacitors and four resistors.In this paper, a new configuration is proposed to realize voltage-mode lowpass, bandpass, and highpass filters simultaneously by using two CFAs, two grounded capacitors, and five resistors. One (or two) more filtering signal can be obtained with respect to the previous two-CFAs biquads in [1, 3, 4]. With respect to the voltage-mode three-CFAs lowpass, bandpass, and highpass biquads in [4–6], the proposed circuit uses one less active component.A new configuration is proposed to realize current-mode lowpass, bandpass and highpass filters simultaneously. One more filtering signal can be obtained with respect to the previous current-mode biquad in [2]. Critical component matching conditions are not required in the design of all proposed circuits.


## 2. Voltage-Mode Filters

Using standard notation, the port relations of a CFA can be characterized byVx=Vy,Vo=Vz,Iz=Ix,andIy=0. The proposed voltage-mode circuit is shown in Figure 1. The output transfer functions of Figure 1 can be expressed as(1)VlpVin=-G1G2G3s2C1C2(G2+G3+G4+G5)+sC1G2G4+G1G2G5,VbpVin=-sC1G2G3s2C1C2(G2+G3+G4+G5)+sC1G2G4+G1G2G5,VhpVin=s2C1C2G3s2C1C2(G2+G3+G4+G5)+sC1G2G4+G1G2G5.Figure 1
The proposed voltage-mode lowpass, bandpass and highpass filter.Thus, the circuit realizes an inverting lowpass signal atVlp, an inverting bandpass signal at Vbp, and a noninverting highpass signal at Vhp, simultaneously. The circuit employs two grounded capacitors, five resistors and only two CFAs. Critical component matching conditions are not required. The output terminals of Vlp and Vbp can be directly connected to the next stage, respectively. The various parameter values of Figure 1 are given by(2)ωo=G1G2G5C1C2(G2+G3+G4+G5),ωoQ=G2G4C2(G2+G3+G4+G5),Q=1G4C2G1G5(G2+G3+G4+G5)C1G2.The gain constants are(3)Ho(lp)=-G3G5,Ho(bp)=-G3G4,Ho(hp)=G3G2+G3+G4+G5.One possible design equations for the specifiedωo and Q can be obtained by(4)G1=Q2G54,C1=C2=QG54ωo,G2=G3=G4=G5.Under the design (4), the gain constants of Figure 1 become(5)Ho(lp)=-1,Ho(bp)=-1Ho(hp)=14.All capacitors are grounded in Figure1. The use of grounded capacitors is particularly attractive for integrated circuit implementation [7]. Moreover, the two capacitors in Figure 1 are connected to the z terminals of the CFAs this design offers another features of (i) a direct incorporation of the parasitic compensation capacitance (Cp) as a part of the main capacitance, and (ii) operation of the proposed circuit in Figure 1 as external-capacitor-less active-R biquad by deleting the external capacitors and accounting only the parasitic compensating capacitors (Cp) into the design [6]. Note that while cascading the highpass signal of Figure 1 to next stage, other buffering device is needed because the output impedance Vhp in Figure 1 is not small.Taking into account the tracking errors of CFAs, namelyVx=β(s)Vy,Vo=γ(s)Vz, and Iz=α(s)Ix, where α(s) and β(s) represent the frequency transfers of the internal current and voltage followers of the CFA, respectively, and γ(s) represents the frequency transfer of the output voltage follower of the CFA. They can be approximated by the first-order lowpass functions [2]. Assume that the circuits are working at frequencies much less than the corner frequencies of α(s), β(s), and γ(s), that is, β=1-ε1 and ε1(|ε1|≪1) is the input voltage tracking error, and  γ=1-ε2 and ε2(|ε2|≪1) is the output voltage tracking error, α=1-ε3 and ε3(|ε3|≪1) is the current tracking error of a CFA. The resonance angular frequency ωo, bandwidth ω0/𝒬, and quality factor Q of Figure 1 become(6)ωo=G1G2G5α1α2β1γ1γ2C1C2(G2+G3+G4+G5),ωoQ=G2G4α2γ2C2(G2+G3+G4+G5),Q=1G4C2G1G5α1β1γ1(G2+G3+G4+G5)C1G2α2γ2.The active and passive sensitivities in Figure1 can be calculated from (6) and found to be less than unity.


## 3. Current-Mode Filters

The proposed current-mode circuit is shown in Figure2. The output transfer functions of Figure 2 can be expressed as(7)IlpIin=-G1G2G6s2C1C2(G2+G3+G4+G5)+sC1G2G4+G1G2G5,IbpIin=-sC1G1G2s2C1C2(G2+G3+G4+G5)+sC1G2G4+G1G2G5,Ihp1Iin=s2C1C2G3s2C1C2(G2+G3+G4+G5)+sC1G2G4+G1G2G5,Ihp2Iin=-s2C1C2G2s2C1C2(G2+G3+G4+G5)+sC1G2G4+G1G2G5.Figure 2
The proposed current-mode filter.Thus, the circuit realizes an inverting lowpass signal atIlp, an inverting bandpass signal at Ibp, a noninverting highpass signal at Ihp1, and an inverting highpass signal at Ihp2, simultaneously. The resonance angular frequency ωo, bandwidth ωo/𝒬, and quality factor Q have the same values as in (6). The gains of Figure 2 are(8)Ho(lp)=-G6G5,Ho(bp)=-G1G4,Ho(hp1)=G3G2+G3+G4+G5Ho(hp2)=-G2G2+G3+G4+G5.Because the output currents are flowing in grounded elements, additional current followers are required for sensing and taking out the output currents. The active and passive sensitivities in Figure2 can be calculated as in voltage-mode circuit and were found to be less than unity.


## 4. Nonideal Equivalent Circuit of CFA

The nonideal equivalent circuit model of the CFA is shown in Figure3, where Rx is the x terminal input resistance, Ry//(1/sCy) represents the y terminal parasitic input impedance, and Rp//(1/sCp) represents the parasitic impedance at the compensation terminal z [2]. The typical data sheet values of the various parasitics for the bipolar CFAs (such as AD844) are Rx = 50Ω, Cp = 5.5 pF, Rp = 3 MΩ, Ry = 2 MΩ and Cy = 2 pF. When non-ideal equivalent circuit model of the CFAs used instead of ideal ones and assuming the circuits are working at frequencies much less than the corner frequencies of α(s), β(s), and γ(s), namely, α≅β≅γ≅1, the voltage transfer functions of Figure 1 become (9)VlpVin=-G1′G2′G3D(s)(10)VbpVin=-(sC1′G2′G3+G2′G3Gp1)D(s),(11)VhpVin=s2C1′C2′G3+sC1′G3Gp2+sC2′G3Gp1+G3Gp1Gp2D(s)Figure 3
Nonideal equivalent circuit of the CFA includes the parasitic impedances.where(12)D(s)=s2C1′C2′(G2′+G3+G4+G5)+s[C1′G2′G4+C1′Gp2(G2′+G3+G4+G5)+G2′Gp1(C2′+G3+G4+G5)]+G1′G2′G5+G2′G4Gp1+Gp1Gp2(G2′+G3+G4+G5),(13)G1′=1/(R1+Rx1);G2′=1/(R2+Rx2);C1′=C1+Cp1;C2′=C2+Cp2From (9) to (11), undesirable factors are yielded by the effects of CFAs’ parasitic impedances. It is found that such factors can be made negligible by operating the filters in high frequencies. But, if the filters are used for lower frequencies, the parasitic impedances could not be negligible. So the characteristics will depart from the theoretical values, especially for the highpass and bandpass filter signals in Figure 1.Note that the influence of the parasitic elements on the frequency responses of the current-mode filters in Figure2 can be studied by a similar procedure, as above.


## 5. Experimental results

Experiments were carried out to demonstrate the feasibility of the proposed circuit. The CFAs were implemented using AD844s. Figures4(a), 4(b), and 4(c) represent the frequency responses for the lowpass, bandpass, and highpass filters of Figure 1, respectively, designed with C1=C2=100pF and R1=R2=R3=R4=R5=10kΩ. Experimental results confirm the theoretical analysis.Experimental frequency responses of Figure1 design with C1=C2=100pFandR1=R2=R3=R4=R5=10kΩ: (a) lowpass filter (Vlp), (b) bandpass filter (Vbp), and (c) highpass filter (Vhp).
(a)(b)(c)


## 6. Conclusions

In this paper, a configuration for realizing voltage-mode multi-function filters and another configuration for realizing current-mode multi-function filters using CFAs are presented. The proposed voltage-mode circuit exhibits simultaneously lowpass, bandpass and highpass filters by using two CFAs, two grounded capacitors, and five resistors. The proposed current-mode circuit exhibits simultaneously lowpass, bandpass, and highpass filters by using two CFAs, two grounded capacitors, and six resistors. The proposed circuits have no requirements for component matching conditions. The active and passive sensitivities are low.



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*Source: 785631-2010-06-10.xml*