Improved Fully Differential Analog Filters - Semantic Scholar
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 6, DECEMBER 2007
Improved Fully Differential Analog Filters Marco Massarotto, Oscar Casas, Member, IEEE, Vittorio Ferrari, Member, IEEE, and Ramon Pallàs-Areny, Fellow, IEEE
Abstract—This paper proposes two novel design techniques that improve the performance of fully differential filters built by coupling two equal single-ended filters. These filters usually reach a high common-mode rejection ratio (CMRR) without using tightly matched passive components but may become unstable because of common-mode signals. Previously proposed techniques to solve these problems, based on resistor or capacitor T-networks, reduce the CMRR. The two techniques proposed overcome this tradeoff. The first technique improves the stability of the filter common-mode gain (GCC ) by adding damping resistors that do not modify the desired differential frequency response (GDD ) or the CMRR. The second technique, which is particularly valuable when a path for input bias currents must be provided, uses a resistor network to estimate the common-mode voltage of the input and uses it as common ground voltage for the filter. Index Terms—Analog filters, circuit stability, common-mode rejection ratio (CMRR), differential circuits, fully differential filters.
I. I NTRODUCTION
D
IFFERENTIAL signaling is commonly used in many fields (audio electronics, data transmission, instrumentation, etc.) mainly because of its inherent immunity to external interference. Furthermore, there is an increasing availability of sensors with differential output and of analog-to-digital converters with differential input. Hence, the design of fully differential analog circuits to directly process the information embedded into a differential signal to achieve a high signal-tonoise ratio is of increasing interest. This paper deals with the design of high-performance fully differential filters, which can be described by four transfer functions [1]: GDD , the desired filtering function; GDC , the common-mode input to differential-mode output gain; GCC , the common-mode input to common-mode output gain; and GCD , the differential-mode input to common-mode output gain. The common-mode rejection ratio (CMRR) is the ratio GDD /GDC . A systematic approach to design fully differential filters, termed “coupling method” [2], consists of connecting singleended filters at their corresponding grounding points and disconnecting the resulting node from ground. This technique avoids grounded passive components from each signal line Manuscript received August 24, 2006; revised April 2, 2007. The works of O. Casas and R. Pallàs-Areny were supported in part by the Spanish Ministry of Science and Technology (MCYT) under Projects TIC2002-03932 and HI03-48, and the work of M. Massarotto was supported in part by the University of Brescia. M. Massarotto and V. Ferrari are with the Faculty of Engineering, University of Brescia, 25133 Brescia, Italy. O. Casas and R. Pallàs-Areny are with the Instrumentation, Sensors, and Interfaces Group, Technical School of Castelldefels, Technical University of Catalonia, 08034 Barcelona, Spain. Digital Object Identifier 10.1109/TIM.2007.904572
to ground, which results in a very high CMRR, regardless of passive-component mismatch. Nevertheless, the resulting circuit must include a path for op-amps’ bias currents. This is a common problem in high-pass circuits and usually ends in added grounded resistor networks that, as a drawback, reduce the CMRR. Furthermore, the design of fully differential circuits should not be limited to obtaining the desired GDD and the highest CMRR possible. Undesirable features, such as instabilities in GDC , GCC , or GCD , or a long transient response of any of the four transfer functions involved, should also be considered [3], [4]. In summary, coupling single-ended stages is a fast method to obtain fully differential filters with high CMRR but does not guarantee circuit stability and needs grounded components to ensure a path for input-bias currents. Solutions proposed in [3] imply a tradeoff between CMRR and stability. This paper proposes two design techniques that simultaneously provide stability and high CMRR and are applied to lowpass, high-pass, and band-pass filters based on the Sallen–Key topologies. II. T HEORETICAL A NALYSIS In fully differential filters, ideally, we wish for GDD to be equal to the desired filtering function GDC = 0 (or CMRR = ∞), GCD = 0, and GCC = 1. In practice, we aim to design filters that, for the desired GDD , yield a high CMRR, are stable, and do not show long transient responses because of the usually disregarded GCC and GDC . We use the coupling method [2] to easily meet the GDD and CMRR design specifications, and we propose two design techniques to meet the common-mode related gains. A. Design Method Based on Compensating the Gain and Phase Margins The first technique relies on adding to the fully differential filters some passive components able to dampen the troublesome frequency response (usually GCC ) without substantially modifying the desired GDD and maintaining a high CMRR. The specific solution, i.e., the site for the added components, depends on the particular filter topology. This technique relies on the general theory to compensate op-amp-based feedback systems that are not quite stable [5]. It basically consists of improving the gain and phase margins of the circuit by modifying the loop gain, e.g., by modifying its passive feedback network. Fig. 1 shows a low-pass second-order fully differential filter built by coupling two equal single-ended Sallen–Key filters.
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Fig. 1. Fully differential low-pass Sallen–Key filter. Uncoupled filter (ncf; CC = ∞, Rd = Rd = 0 Ω); coupled filter (cf; CC = 0 F, Rd = Rd = 0 Ω); coupled filter stabilized by CC (Rd = Rd = 0 Ω); proposed coupled filter stabilized by Rd , Rd (CC = ∞).
Reference [3] shows that this topology is prone to oscillation due to GCC and proposes to stabilize it by a small capacitance CC from the node P to ground; this solution, however, leads to a tradeoff between stability and CMRR. We propose to add, instead, two equal stabilization resistors Rd and Rd , as shown in Fig. 1. This technique does not involve any direct connection between signal lines and ground (C2 and C2 can be replaced by their series equivalent) so that we fully exploit the advantages of the coupling method. Consequently, we achieve a high CMRR, regardless of passive-component mismatch. In Fig. 1, if we initially assume matched passive components (Rd = Rd , Ri = Ri , Ci = Ci ; i = 1, 2) and ideal op amps, we obtain GDC = GCD = 0 (CMRR = ∞), GCC = 1, and (1), shown at the bottom of the page. Rd has only a minor effect on GDD , provided that Rd R1 R2 and Rd C1 ω 1. In addition, because GDD does not include any subtraction, passivecomponent mismatches are irrelevant. Furthermore, if stray capacitances Cin from each op-amp noninverting input to ground are considered, GCC becomes (2), shown at the bottom of the page, whose damping factor is Cin (R1 + R2 ) + C1 Rd . ζ= 2 Cin C1 (R1 R2 + Rd (R1 + R2 ))
(3)
When Rd = 0, ζ can become small enough for GCC to have a large resonance peak, and the filter would oscillate, thus becoming useless, regardless of how large the actual CMRR may be. Solving for Rd in (3), with Rd R1 R2 , yields Cin Cin Rd = Rd ≈ 2ζ R1 R2 − (R1 + R2 ) (4) C1 C1 √ which allows us to design Rd for a target ζ (e.g., ζ = 2/2). If C1 Cin , then relatively small values of Rd can stabilize the filter without significantly modifying GDD . Stray capacitances at op-amp inputs also provide an undesired path to ground for common-mode signals to yield a differential output so that, in practice, GDC is not zero, and the CMRR is limited by the mismatch of passive components. If capacitors (other than Cin ) have a tolerance β and resistors have a tolerance α, the CMRR is (5), shown at the bottom of the page. Hence, in order to minimize the effect of Cin on CMRR and GDD , we must select C1 , C2 Cin . From (5), for Cin = 0, the CMRR is infinite, regardless of the mismatch of passive components. For a correct design, the gain-bandwidth product fT of op amps must be larger than the filter gain at the corner (or any intermediate) frequency. Moreover, op-amp mismatches will limit the CMRR. For Cin = 0 and matched passive components, the influence of op-amp mismatch on the overall CMRR is 1 1 1 1 1 = − + − CMRR Ad1 Ad2 CMRR1 CMRR2
(6)
where Adi are the open-loop gains, and CMRRi are the CMRRs of the two op amps [12]. In practice, this last drawback mainly has effects at high frequencies and can be partially mitigated by using a fully differential op amp [6] instead of a pair of op amps with single-ended output. B. Design Method Based on Estimating the Common-Mode Voltage The second design technique that we propose is shown in Fig. 2. For R = R = RC = ∞, the circuit would be a
GDD =
Rd C1 s + 1 [R1 R2 + Rd (R1 + R2 )] C1 C2 s2 + [(R1 + R2 )C2 + Rd C1 ] s + 1
(1)
GCC =
Rd C1 s + 1 [R1 R2 + Rd (R1 + R2 )] C1 Cin s2 + [(R1 + R2 )Cin + Rd C1 ] s + 1
(2)
[(R1 + R2 )C2 s + 1] (R1 + R2 )C1 Cin s2 + (R1 + R2 )Cin s + 1 CMRR ≈ 2s(R1 + R2 )Cin (α + β) [R1 R2 C1 C2 s2 + (R1 + R2 )C2 s + 1]
(5)
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Fig. 2. Fully differential high-pass Sallen–Key filter. Uncoupled filter (ncf; RC = 0 Ω; R = R = ∞); coupled filter (cf; RC = ∞; R = R = ∞) without bias path for op amps (hence, it would not work); coupled filter with op amps biased through RC (R = R = ∞); coupled filter with op amps biased through R = R (RC = ∞).
high-pass second-order fully differential filter built by the coupling method. For the circuit to operate properly, a path should be provided for bias currents. Adding a large resistor RC from P to ground provides such a path and keeps the CMRR high [2], but the circuit can display a long transient response because of GDC [3] and, worse yet, be unstable because of GCC . The solution that we propose consists of the following: 1) omitting RC (RC = ∞); 2) estimating the input commonmode voltage νC (node N ) by two equal resistors R and R ; and 3) connecting N to P . Then, in Fig. 2, we obtain
R R + R R = VC − VD /2 + VD ≈ VC . R + R
VN,P = VL + VD
(7)
It has been assumed that the internal impedance of the signal source is resistive and that R and R are much larger than the impedance. Then, for ideal op amps and matched components, we obtain GDC = GCD = 0, GCC = 1, and GDD =
R1 R2 C1 C2 s2 . R1 R2 C1 C2 s2 + R1 (C1 + C2 )s + 1
(8)
If passive components are mismatched, GDD and GCD become slightly different, but we still have GCC = 1 and GDC = 0 (CMRR = ∞). This is because there is no way a commonmode input voltage can produce a differential-mode output voltage. Therefore, the inclusion of R and R provides a bias path, does not modify the original GDD , yields a theoretically infinite CMRR, and keeps GCC = 1, and hence, stability, irrespective of passive component tolerance. Stray capacitance to ground and op-amp mismatch limit the maximal CMRR. This design method relies on the capability to estimate the common-mode voltage of the input signal and to connect it to the common ground of the filter and, at the same time, providing a bias path for op amps. Therefore, this is a general solution that can be applied to other circuits. Consider, for example, the
Fig. 3. Fully differential band-pass filter built by coupling two equal Sallen–Key single-ended filters, whose common ground is connected to the common-mode input voltage (R and R serve to estimate the common-mode).
second-order band-pass fully differential filter in Fig. 3. Adding R and R provides a bias path yet avoids any connections to ground, thus resulting in a high CMRR. R4 and R4 can be replaced by their series equivalent but not R2 and R2 . Another example is the high-pass passive fully differential filter in [7]. III. M ATERIALS AND M ETHODS To validate the two design methods proposed, we have designed three different filter types and tested them with the measurement setup shown in Fig. 4. We have measured GDD , GDC , and GCC and calculated the CMRR as CMRR = GDD /GDC . The input stimulus was a sinusoidal voltage applied either in differential mode (for GDD ) or in common mode (for GDC and GCC ). The differential-mode input signal was obtained from a single-ended function generator by a buffer-inverter circuit [9], [10]. The symmetry of the two components of the differential signal was improved, and the effects of the opamp offset currents were limited by using two wideband op amps (EL2244) and R = 100 Ω (1%). The load seen by the function generator (R/2 = 50 Ω) was matched to its output impedance. The instrumentation amplifier (IA) in Fig. 4 provides an amplified single-ended output to the measuring instrument. We used the INA110 set to gain 200 (100-kHz bandwidth), whose low-frequency CMRR is 118 dB (corner frequency, 1 kHz). The overall CMRR is [9]–[12] 1 1 1 ≈ + . CMRR CMRRFilter CMRRINA Therefore, as long as CMRRFilter CMRRINA , the measured CMRR will be that of the filter. To obtain GDD , GDC , and GCC , the input and output rms voltages were measured with two voltmeters: Prema 5017 (7 1/2 digits) and Agilent 3478A (5 1/2 digits). The measurement process was automated by LabVIEW and monitored by an oscilloscope in order to detect any instability or long time
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Fig. 4. Measurement system we used to test the presented fully differential filters. It consists of sinusoidal generator, a common- and differential-mode signal generator, a test-signal selector, a filter under test, and an IA.
transients. A simple algorithm adaptively adjusted the input signal amplitude to optimally use the dynamic range of the output voltmeter. The power supply (±15 V) had a negligible influence on the measurement because of its low-output rms noise (500 µV), the use of power supply decoupling capacitors, and the powersupply rejection ratio of op amps. All the filters tested were designed to provide unity gain and 100-Hz corner (low-pass, high-pass) or center (band-pass) frequency. Tolerances were 5% for resistors and 20% for capacitors. Op amps were off-theshelf TL072.
IV. E XPERIMENTAL R ESULTS AND D ISCUSSION We have built the low-pass filter in Fig. 1 by using the following component values: R1 = R1 = 15 kΩ, C1 = C1 = 100 nF, R2 = R2 = 56 kΩ, and C2 = C2 = 33 nF. Fig. 5 shows the experimental |GDD |, |CMRR|, and |GCC | versus frequency for four different circuit versions: 1) ncf, the uncoupled filter (CC = ∞, Rd = Rd = 0); 2) cf, the basic coupled filter (CC = 0, Rd = Rd = 0); 3) the coupled filter stabilized by CC (CC = 10 nF, 220 pF; Rd = Rd = 0); and 4) the proposed coupled filter, stabilized by damping resistors Rd , Rd (Rd = Rd = 240 Ω; CC = ∞). GDC is not shown because it does not add any further information. We observe that |GDD | is about 46 dB, due to the gain of the IA, and that its pass band is slightly narrowed because of Rd and Rd . The CMRR is quite poor for the noncoupled filter, and it is notably better for the three coupled filters. The CMRR, which is ideally infinite, has, in practice, a finite value that agrees with the theoretical prediction (5). The gain GCC shows no problems for the noncoupled filter, but it displays a resonance peak for the coupled circuits [Fig. 5(c)]. For the basic coupled filter, the resonance happens in the stop band and could go unnoticed. Adding CC brings the resonance frequency closer to the pass band (an undesired feature) and reduces the resonance amplitude (a desired feature). The larger CC , the stronger the damping, but also, the closer the GCC resonance to the pass band, the worse the CMRR. A largeenough Rd reduces the amplitude of the resonance in GCC but keeps its frequency in the stop band and, furthermore, yields the same high CMRR as the basic coupled filter. This meets √ our theoretical predictions. From (4), in order to achieve ζ = 2/2
Fig. 5. (a) |GDD |, (b) |CMRR|, and (c) |GCC | versus frequency for the lowpass filter of Fig. 1. We selected a corner frequency of 100 Hz. We tested four different topologies obtained by properly changing the values of Rd = Rd and CC .
with the current design parameters and assuming Cin = 6 pF, we needed Rd = Rd = 240 Ω. However, Rd = 240 Ω did not result in a flat frequency response GCC , which suggests that Cin was probably a bit larger than 6 pF. We have built the high-pass filter in Fig. 2 by using R1 = R1 = 15 kΩ, C1 = C1 = 100 nF, R2 = R2 = 47 kΩ, and
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Fig. 6. (a) |GDD |, (b) |CMRR|, and (c) |GCC | versus frequency for the highpass filter in Fig. 2. We selected a corner frequency of 100 Hz and tested two different topologies obtained by using either RC = 1 MΩ(Rd = Rd = ∞) or Rd = Rd = 1 MΩ(RC = ∞).
C2 = C2 = 33 nF. Fig. 6 shows |GDD |, |CMRR|, and |GCC | versus frequency for two different circuits: 1) that is based on a resistor T-network (RC = 1 MΩ; R = R = ∞) and 2) the proposed circuit (R = R = 1 MΩ; RC = ∞). Both circuits yield the desired GDD , but GCC has a resonance peak in the first circuit, yet it is flat in the second circuit, which, in addition, has higher CMRR. Note that, the same as for the filter in Fig. 1, the resonance peak in GCC falls in the stop band so that it may go unnoticed. We have also checked that a smaller RC reduces the resonance peak in the first circuit, as well as its CMRR [3], and that larger R and R values for the second circuit did not improve its CMRR. Finally, we have built the band-pass filter in Fig. 3 by using R1 = R1 = 33 kΩ, R2 = R2 = 20 kΩ, R3 = R3 = 15 kΩ, R4 = R4 = R5 = R5 = 47 kΩ, and C1 = C1 = C2 = C2 = 100 nF. We have tested a circuit that included R = R = 1 MΩ. Fig. 7 shows the corresponding |GDD |, |CMRR|, and |GCC | versus frequency. The filter has the desired |GDD | and |CMRR| of about 70 dB in the band pass, which is remarkable for a unity-gain filter based on 5% tolerance resistors, and a |GCC |
Fig. 7. (a) |GDD |, (b) |CMRR|, and (c) |GCC | versus frequency for the bandpass filter of Fig. 3. We selected a central frequency of 100 Hz, and R = R = 1 MΩ.
that is flat and close to the ideal unity value. These results reassert the validity of the design technique proposed. V. C ONCLUSION Fully differential high-performance filters are becoming increasingly important in instrumentation electronics, but no systematic method is yet available to design them without using tightly matched components. By coupling single-ended filter stages, as proposed in [2], it is easy to meet the required filtering function GDD and to obtain a high CMRR, but at the risk of some common-mode-related problems because of GCC or GDC . Therefore, in spite of its simplicity, its use has been somewhat discouraged [4]. In this paper, we have proposed two design techniques that are able to improve the performance of fully differential filters based on coupled single-ended filters. The first technique consists of adding damping resistors to the filter network in an appropriate position. We have shown how to proceed for a low-pass filter, and we have estimated the minimal value of the resistors to achieve a given damping factor for GCC
MASSAROTTO et al.: IMPROVED FULLY DIFFERENTIAL ANALOG FILTERS
when considering stray capacitances from noninverting op-amp inputs to ground [1]. The second design technique relies on estimating the common-mode voltage of the input signal by a simple resistor network and using that voltage as the common reference voltage for the coupled single-ended filters. This method is particularly interesting for high-pass filters because it provides a bias path for the op-amp inputs, but it can also be applied to other filter types, such as band-pass filters. Both design methods proposed yield the desired frequency response, stable response to common-mode voltages, and high CMRR. There is no tradeoff between stability and CMRR, which happens in circuits that rely on resistor or capacitor T-networks [2], [3].
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Oscar Casas (S’93–M’99) received the Ingeniero de Telecomunicación and Doctor Ingeniero de Telecomunicación degrees from the Universitat Politècnica de Catalunya, Barcelona, Spain, in 1994 and 1998, respectively. He is currently an Associate Professor of electronic engineering with the Instrumentation, Sensors, and Interfaces Group, Technical School of Castelldefels, Technical University of Catalonia, Barcelona.Where he teaches courses in several areas of electronic instrumentation. His research includes sensor interfaces, autonomous sensors, electronic instrumentation, noninvasive physiological measurements, and sensors based on electrical-impedance measurements.
R EFERENCES [1] R. Pallàs-Areny and J. G. Webster, Analog Signal Processing. New York: Wiley, 1999, ch. 2. [2] O. Casas and R. Pallàs-Areny, “Basics of analog differential filters,” IEEE Trans. Instrum. Meas., vol. 45, no. 1, pp. 275–279, Feb. 1996. [3] M. Gasulla, O. Casas, and R. Pallàs-Areny, “On the common mode response of fully differential circuits,” in Proc. 17th IEEE IMTC, 2000, vol. 2, pp. 1045–1049. [4] J. M. da Cunha, “A compact and flexible signal conditioning system for data acquisition,” Hewlett Packard J., vol. 45, no. 5, pp. 9–15, Oct. 1994. [5] S. Franco, Design With Operational Amplifiers and Analog Integrated Circuits. New York: McGraw-Hill, 2002. [6] J. Karki, “Fully differential amplifier,” Texas Instruments, Dallas, TX, Application Rep. SLOA054D, 2002. [7] E. M. Spinelli, R. Pallàs-Areny, and M. A. Mayosky, “AC-coupled frontend for biopotential measurements,” IEEE Trans. Biomed. Eng., vol. 50, no. 3, pp. 391–395, Mar. 2003. [8] M. A. Al-Alaoui, “A differential integrator with a built-in high-frequency compensation,” IEEE Trans. Circuits Syst. I, vol. 45, no. 5, pp. 517–522, May 1998. [9] H. Golnabi and A. Ashrafi, “Producing 180◦ out-of-phase signals from a sinusoidal waveform input,” IEEE Trans. Instrum. Meas., vol. 45, no. 1, pp. 312–314, Feb. 1996. [10] R. Casas, O. Casas, and V. Ferrari, “Single-ended input to differential output circuits. A comparative analysis,” in Proc. 23th IEEE IMTC, 2006, pp. 548–551. [11] R. Pallás-Areny and J. G. Webster, “Common mode rejection ratio for cascaded differential amplifier stages,” IEEE Trans. Instrum. Meas., vol. 40, no. 4, pp. 677–681, Aug. 1991. [12] R. Pallás-Areny and J. G. Webster, “Common mode rejection ratio in differential amplifiers,” IEEE Trans. Instrum. Meas., vol. 40, no. 4, pp. 669–676, Aug. 1991.
Marco Massarotto received the degree in electronic engineering and the Ph.D. degree in electronic instrumentation from the University of Brescia, Brescia, Italy, in 2003 and 2007, respectively. Within the electronic instrumentation and measurements fields, his main researches regard differential circuits (voltage filters and charge amplifiers) and the characterization of accelerometers.
Vittorio Ferrari (M’04) was born in Milan, Italy, in 1962. He received the Laurea degree in physics (cum laude) from the University of Milan in 1988 and the Research Doctorate degree in electronic instrumentation from the University of Brescia, Brescia, Italy, in 1993. Since 2006, he has been a Full Professor of electronics with the Faculty of Engineering, University of Brescia. His research activity is in the field of sensors and the related signal-conditioning electronics. Topics of interest are acoustic-wave piezoelectric sensors, microresonant sensors and microelectromechanical systems, autonomous sensors and power scavenging, oscillators for resonant sensors, and frequency-output interface circuits. He is involved with national and international research programs and with projects in cooperation with industries.
Ramon Pallàs-Areny (M’81–SM’88–F’98) received the Ingeniero Industrial and Doctor Ingeniero Industrial degrees from the Technical University of Catalonia (UPC), Barcelona, Spain, in 1975 and 1982, respectively. From 1989 to 1990, he was a Visiting Fulbright Scholar, and from 1997 to 1998, he was an Honorary Fellow with the University of Wisconsin, Madison. In 2001, he was nominated as Professor Honoris Causa by the Faculty of Electrical Engineering, University of Cluj-Napoca, Romania. He is currently a Professor of electronic engineering with UPC and teaches courses in electronic instrumentation. His research includes instrumentation methods and sensors based on electrical-impedance measurements, autonomous sensors, sensor interfaces, noninvasive physiological measurements, and electromagnetic compatibility in electronic systems. He is the author of six books, the leading author of five books, and coauthor of two books on instrumentation in Spanish and Catalan. He is also the coauthor (with J. G. Webster) of Sensors and Signal Conditioning, 2nd ed. (New York: Wiley, 2001) and Analog Signal Processing (New York: Wiley, 1999). Dr. Pallàs-Areny was the recipient, with J. G. Webster, of the 1991 Andrew R. Chi Prize Paper Award from the IEEE Instrumentation and Measurement Society. In 2000, he was the recipient of the Award for Quality in Teaching granted by the Board of Trustees of UPC and, in 2002, the Narcís Monturiol Medal from the Autonomous Government of Catalonia.
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 6, DECEMBER 2007
Improved Fully Differential Analog Filters Marco Massarotto, Oscar Casas, Member, IEEE, Vittorio Ferrari, Member, IEEE, and Ramon Pallàs-Areny, Fellow, IEEE
Abstract—This paper proposes two novel design techniques that improve the performance of fully differential filters built by coupling two equal single-ended filters. These filters usually reach a high common-mode rejection ratio (CMRR) without using tightly matched passive components but may become unstable because of common-mode signals. Previously proposed techniques to solve these problems, based on resistor or capacitor T-networks, reduce the CMRR. The two techniques proposed overcome this tradeoff. The first technique improves the stability of the filter common-mode gain (GCC ) by adding damping resistors that do not modify the desired differential frequency response (GDD ) or the CMRR. The second technique, which is particularly valuable when a path for input bias currents must be provided, uses a resistor network to estimate the common-mode voltage of the input and uses it as common ground voltage for the filter. Index Terms—Analog filters, circuit stability, common-mode rejection ratio (CMRR), differential circuits, fully differential filters.
I. I NTRODUCTION
D
IFFERENTIAL signaling is commonly used in many fields (audio electronics, data transmission, instrumentation, etc.) mainly because of its inherent immunity to external interference. Furthermore, there is an increasing availability of sensors with differential output and of analog-to-digital converters with differential input. Hence, the design of fully differential analog circuits to directly process the information embedded into a differential signal to achieve a high signal-tonoise ratio is of increasing interest. This paper deals with the design of high-performance fully differential filters, which can be described by four transfer functions [1]: GDD , the desired filtering function; GDC , the common-mode input to differential-mode output gain; GCC , the common-mode input to common-mode output gain; and GCD , the differential-mode input to common-mode output gain. The common-mode rejection ratio (CMRR) is the ratio GDD /GDC . A systematic approach to design fully differential filters, termed “coupling method” [2], consists of connecting singleended filters at their corresponding grounding points and disconnecting the resulting node from ground. This technique avoids grounded passive components from each signal line Manuscript received August 24, 2006; revised April 2, 2007. The works of O. Casas and R. Pallàs-Areny were supported in part by the Spanish Ministry of Science and Technology (MCYT) under Projects TIC2002-03932 and HI03-48, and the work of M. Massarotto was supported in part by the University of Brescia. M. Massarotto and V. Ferrari are with the Faculty of Engineering, University of Brescia, 25133 Brescia, Italy. O. Casas and R. Pallàs-Areny are with the Instrumentation, Sensors, and Interfaces Group, Technical School of Castelldefels, Technical University of Catalonia, 08034 Barcelona, Spain. Digital Object Identifier 10.1109/TIM.2007.904572
to ground, which results in a very high CMRR, regardless of passive-component mismatch. Nevertheless, the resulting circuit must include a path for op-amps’ bias currents. This is a common problem in high-pass circuits and usually ends in added grounded resistor networks that, as a drawback, reduce the CMRR. Furthermore, the design of fully differential circuits should not be limited to obtaining the desired GDD and the highest CMRR possible. Undesirable features, such as instabilities in GDC , GCC , or GCD , or a long transient response of any of the four transfer functions involved, should also be considered [3], [4]. In summary, coupling single-ended stages is a fast method to obtain fully differential filters with high CMRR but does not guarantee circuit stability and needs grounded components to ensure a path for input-bias currents. Solutions proposed in [3] imply a tradeoff between CMRR and stability. This paper proposes two design techniques that simultaneously provide stability and high CMRR and are applied to lowpass, high-pass, and band-pass filters based on the Sallen–Key topologies. II. T HEORETICAL A NALYSIS In fully differential filters, ideally, we wish for GDD to be equal to the desired filtering function GDC = 0 (or CMRR = ∞), GCD = 0, and GCC = 1. In practice, we aim to design filters that, for the desired GDD , yield a high CMRR, are stable, and do not show long transient responses because of the usually disregarded GCC and GDC . We use the coupling method [2] to easily meet the GDD and CMRR design specifications, and we propose two design techniques to meet the common-mode related gains. A. Design Method Based on Compensating the Gain and Phase Margins The first technique relies on adding to the fully differential filters some passive components able to dampen the troublesome frequency response (usually GCC ) without substantially modifying the desired GDD and maintaining a high CMRR. The specific solution, i.e., the site for the added components, depends on the particular filter topology. This technique relies on the general theory to compensate op-amp-based feedback systems that are not quite stable [5]. It basically consists of improving the gain and phase margins of the circuit by modifying the loop gain, e.g., by modifying its passive feedback network. Fig. 1 shows a low-pass second-order fully differential filter built by coupling two equal single-ended Sallen–Key filters.
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Fig. 1. Fully differential low-pass Sallen–Key filter. Uncoupled filter (ncf; CC = ∞, Rd = Rd = 0 Ω); coupled filter (cf; CC = 0 F, Rd = Rd = 0 Ω); coupled filter stabilized by CC (Rd = Rd = 0 Ω); proposed coupled filter stabilized by Rd , Rd (CC = ∞).
Reference [3] shows that this topology is prone to oscillation due to GCC and proposes to stabilize it by a small capacitance CC from the node P to ground; this solution, however, leads to a tradeoff between stability and CMRR. We propose to add, instead, two equal stabilization resistors Rd and Rd , as shown in Fig. 1. This technique does not involve any direct connection between signal lines and ground (C2 and C2 can be replaced by their series equivalent) so that we fully exploit the advantages of the coupling method. Consequently, we achieve a high CMRR, regardless of passive-component mismatch. In Fig. 1, if we initially assume matched passive components (Rd = Rd , Ri = Ri , Ci = Ci ; i = 1, 2) and ideal op amps, we obtain GDC = GCD = 0 (CMRR = ∞), GCC = 1, and (1), shown at the bottom of the page. Rd has only a minor effect on GDD , provided that Rd R1 R2 and Rd C1 ω 1. In addition, because GDD does not include any subtraction, passivecomponent mismatches are irrelevant. Furthermore, if stray capacitances Cin from each op-amp noninverting input to ground are considered, GCC becomes (2), shown at the bottom of the page, whose damping factor is Cin (R1 + R2 ) + C1 Rd . ζ= 2 Cin C1 (R1 R2 + Rd (R1 + R2 ))
(3)
When Rd = 0, ζ can become small enough for GCC to have a large resonance peak, and the filter would oscillate, thus becoming useless, regardless of how large the actual CMRR may be. Solving for Rd in (3), with Rd R1 R2 , yields Cin Cin Rd = Rd ≈ 2ζ R1 R2 − (R1 + R2 ) (4) C1 C1 √ which allows us to design Rd for a target ζ (e.g., ζ = 2/2). If C1 Cin , then relatively small values of Rd can stabilize the filter without significantly modifying GDD . Stray capacitances at op-amp inputs also provide an undesired path to ground for common-mode signals to yield a differential output so that, in practice, GDC is not zero, and the CMRR is limited by the mismatch of passive components. If capacitors (other than Cin ) have a tolerance β and resistors have a tolerance α, the CMRR is (5), shown at the bottom of the page. Hence, in order to minimize the effect of Cin on CMRR and GDD , we must select C1 , C2 Cin . From (5), for Cin = 0, the CMRR is infinite, regardless of the mismatch of passive components. For a correct design, the gain-bandwidth product fT of op amps must be larger than the filter gain at the corner (or any intermediate) frequency. Moreover, op-amp mismatches will limit the CMRR. For Cin = 0 and matched passive components, the influence of op-amp mismatch on the overall CMRR is 1 1 1 1 1 = − + − CMRR Ad1 Ad2 CMRR1 CMRR2
(6)
where Adi are the open-loop gains, and CMRRi are the CMRRs of the two op amps [12]. In practice, this last drawback mainly has effects at high frequencies and can be partially mitigated by using a fully differential op amp [6] instead of a pair of op amps with single-ended output. B. Design Method Based on Estimating the Common-Mode Voltage The second design technique that we propose is shown in Fig. 2. For R = R = RC = ∞, the circuit would be a
GDD =
Rd C1 s + 1 [R1 R2 + Rd (R1 + R2 )] C1 C2 s2 + [(R1 + R2 )C2 + Rd C1 ] s + 1
(1)
GCC =
Rd C1 s + 1 [R1 R2 + Rd (R1 + R2 )] C1 Cin s2 + [(R1 + R2 )Cin + Rd C1 ] s + 1
(2)
[(R1 + R2 )C2 s + 1] (R1 + R2 )C1 Cin s2 + (R1 + R2 )Cin s + 1 CMRR ≈ 2s(R1 + R2 )Cin (α + β) [R1 R2 C1 C2 s2 + (R1 + R2 )C2 s + 1]
(5)
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Fig. 2. Fully differential high-pass Sallen–Key filter. Uncoupled filter (ncf; RC = 0 Ω; R = R = ∞); coupled filter (cf; RC = ∞; R = R = ∞) without bias path for op amps (hence, it would not work); coupled filter with op amps biased through RC (R = R = ∞); coupled filter with op amps biased through R = R (RC = ∞).
high-pass second-order fully differential filter built by the coupling method. For the circuit to operate properly, a path should be provided for bias currents. Adding a large resistor RC from P to ground provides such a path and keeps the CMRR high [2], but the circuit can display a long transient response because of GDC [3] and, worse yet, be unstable because of GCC . The solution that we propose consists of the following: 1) omitting RC (RC = ∞); 2) estimating the input commonmode voltage νC (node N ) by two equal resistors R and R ; and 3) connecting N to P . Then, in Fig. 2, we obtain
R R + R R = VC − VD /2 + VD ≈ VC . R + R
VN,P = VL + VD
(7)
It has been assumed that the internal impedance of the signal source is resistive and that R and R are much larger than the impedance. Then, for ideal op amps and matched components, we obtain GDC = GCD = 0, GCC = 1, and GDD =
R1 R2 C1 C2 s2 . R1 R2 C1 C2 s2 + R1 (C1 + C2 )s + 1
(8)
If passive components are mismatched, GDD and GCD become slightly different, but we still have GCC = 1 and GDC = 0 (CMRR = ∞). This is because there is no way a commonmode input voltage can produce a differential-mode output voltage. Therefore, the inclusion of R and R provides a bias path, does not modify the original GDD , yields a theoretically infinite CMRR, and keeps GCC = 1, and hence, stability, irrespective of passive component tolerance. Stray capacitance to ground and op-amp mismatch limit the maximal CMRR. This design method relies on the capability to estimate the common-mode voltage of the input signal and to connect it to the common ground of the filter and, at the same time, providing a bias path for op amps. Therefore, this is a general solution that can be applied to other circuits. Consider, for example, the
Fig. 3. Fully differential band-pass filter built by coupling two equal Sallen–Key single-ended filters, whose common ground is connected to the common-mode input voltage (R and R serve to estimate the common-mode).
second-order band-pass fully differential filter in Fig. 3. Adding R and R provides a bias path yet avoids any connections to ground, thus resulting in a high CMRR. R4 and R4 can be replaced by their series equivalent but not R2 and R2 . Another example is the high-pass passive fully differential filter in [7]. III. M ATERIALS AND M ETHODS To validate the two design methods proposed, we have designed three different filter types and tested them with the measurement setup shown in Fig. 4. We have measured GDD , GDC , and GCC and calculated the CMRR as CMRR = GDD /GDC . The input stimulus was a sinusoidal voltage applied either in differential mode (for GDD ) or in common mode (for GDC and GCC ). The differential-mode input signal was obtained from a single-ended function generator by a buffer-inverter circuit [9], [10]. The symmetry of the two components of the differential signal was improved, and the effects of the opamp offset currents were limited by using two wideband op amps (EL2244) and R = 100 Ω (1%). The load seen by the function generator (R/2 = 50 Ω) was matched to its output impedance. The instrumentation amplifier (IA) in Fig. 4 provides an amplified single-ended output to the measuring instrument. We used the INA110 set to gain 200 (100-kHz bandwidth), whose low-frequency CMRR is 118 dB (corner frequency, 1 kHz). The overall CMRR is [9]–[12] 1 1 1 ≈ + . CMRR CMRRFilter CMRRINA Therefore, as long as CMRRFilter CMRRINA , the measured CMRR will be that of the filter. To obtain GDD , GDC , and GCC , the input and output rms voltages were measured with two voltmeters: Prema 5017 (7 1/2 digits) and Agilent 3478A (5 1/2 digits). The measurement process was automated by LabVIEW and monitored by an oscilloscope in order to detect any instability or long time
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Fig. 4. Measurement system we used to test the presented fully differential filters. It consists of sinusoidal generator, a common- and differential-mode signal generator, a test-signal selector, a filter under test, and an IA.
transients. A simple algorithm adaptively adjusted the input signal amplitude to optimally use the dynamic range of the output voltmeter. The power supply (±15 V) had a negligible influence on the measurement because of its low-output rms noise (500 µV), the use of power supply decoupling capacitors, and the powersupply rejection ratio of op amps. All the filters tested were designed to provide unity gain and 100-Hz corner (low-pass, high-pass) or center (band-pass) frequency. Tolerances were 5% for resistors and 20% for capacitors. Op amps were off-theshelf TL072.
IV. E XPERIMENTAL R ESULTS AND D ISCUSSION We have built the low-pass filter in Fig. 1 by using the following component values: R1 = R1 = 15 kΩ, C1 = C1 = 100 nF, R2 = R2 = 56 kΩ, and C2 = C2 = 33 nF. Fig. 5 shows the experimental |GDD |, |CMRR|, and |GCC | versus frequency for four different circuit versions: 1) ncf, the uncoupled filter (CC = ∞, Rd = Rd = 0); 2) cf, the basic coupled filter (CC = 0, Rd = Rd = 0); 3) the coupled filter stabilized by CC (CC = 10 nF, 220 pF; Rd = Rd = 0); and 4) the proposed coupled filter, stabilized by damping resistors Rd , Rd (Rd = Rd = 240 Ω; CC = ∞). GDC is not shown because it does not add any further information. We observe that |GDD | is about 46 dB, due to the gain of the IA, and that its pass band is slightly narrowed because of Rd and Rd . The CMRR is quite poor for the noncoupled filter, and it is notably better for the three coupled filters. The CMRR, which is ideally infinite, has, in practice, a finite value that agrees with the theoretical prediction (5). The gain GCC shows no problems for the noncoupled filter, but it displays a resonance peak for the coupled circuits [Fig. 5(c)]. For the basic coupled filter, the resonance happens in the stop band and could go unnoticed. Adding CC brings the resonance frequency closer to the pass band (an undesired feature) and reduces the resonance amplitude (a desired feature). The larger CC , the stronger the damping, but also, the closer the GCC resonance to the pass band, the worse the CMRR. A largeenough Rd reduces the amplitude of the resonance in GCC but keeps its frequency in the stop band and, furthermore, yields the same high CMRR as the basic coupled filter. This meets √ our theoretical predictions. From (4), in order to achieve ζ = 2/2
Fig. 5. (a) |GDD |, (b) |CMRR|, and (c) |GCC | versus frequency for the lowpass filter of Fig. 1. We selected a corner frequency of 100 Hz. We tested four different topologies obtained by properly changing the values of Rd = Rd and CC .
with the current design parameters and assuming Cin = 6 pF, we needed Rd = Rd = 240 Ω. However, Rd = 240 Ω did not result in a flat frequency response GCC , which suggests that Cin was probably a bit larger than 6 pF. We have built the high-pass filter in Fig. 2 by using R1 = R1 = 15 kΩ, C1 = C1 = 100 nF, R2 = R2 = 47 kΩ, and
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Fig. 6. (a) |GDD |, (b) |CMRR|, and (c) |GCC | versus frequency for the highpass filter in Fig. 2. We selected a corner frequency of 100 Hz and tested two different topologies obtained by using either RC = 1 MΩ(Rd = Rd = ∞) or Rd = Rd = 1 MΩ(RC = ∞).
C2 = C2 = 33 nF. Fig. 6 shows |GDD |, |CMRR|, and |GCC | versus frequency for two different circuits: 1) that is based on a resistor T-network (RC = 1 MΩ; R = R = ∞) and 2) the proposed circuit (R = R = 1 MΩ; RC = ∞). Both circuits yield the desired GDD , but GCC has a resonance peak in the first circuit, yet it is flat in the second circuit, which, in addition, has higher CMRR. Note that, the same as for the filter in Fig. 1, the resonance peak in GCC falls in the stop band so that it may go unnoticed. We have also checked that a smaller RC reduces the resonance peak in the first circuit, as well as its CMRR [3], and that larger R and R values for the second circuit did not improve its CMRR. Finally, we have built the band-pass filter in Fig. 3 by using R1 = R1 = 33 kΩ, R2 = R2 = 20 kΩ, R3 = R3 = 15 kΩ, R4 = R4 = R5 = R5 = 47 kΩ, and C1 = C1 = C2 = C2 = 100 nF. We have tested a circuit that included R = R = 1 MΩ. Fig. 7 shows the corresponding |GDD |, |CMRR|, and |GCC | versus frequency. The filter has the desired |GDD | and |CMRR| of about 70 dB in the band pass, which is remarkable for a unity-gain filter based on 5% tolerance resistors, and a |GCC |
Fig. 7. (a) |GDD |, (b) |CMRR|, and (c) |GCC | versus frequency for the bandpass filter of Fig. 3. We selected a central frequency of 100 Hz, and R = R = 1 MΩ.
that is flat and close to the ideal unity value. These results reassert the validity of the design technique proposed. V. C ONCLUSION Fully differential high-performance filters are becoming increasingly important in instrumentation electronics, but no systematic method is yet available to design them without using tightly matched components. By coupling single-ended filter stages, as proposed in [2], it is easy to meet the required filtering function GDD and to obtain a high CMRR, but at the risk of some common-mode-related problems because of GCC or GDC . Therefore, in spite of its simplicity, its use has been somewhat discouraged [4]. In this paper, we have proposed two design techniques that are able to improve the performance of fully differential filters based on coupled single-ended filters. The first technique consists of adding damping resistors to the filter network in an appropriate position. We have shown how to proceed for a low-pass filter, and we have estimated the minimal value of the resistors to achieve a given damping factor for GCC
MASSAROTTO et al.: IMPROVED FULLY DIFFERENTIAL ANALOG FILTERS
when considering stray capacitances from noninverting op-amp inputs to ground [1]. The second design technique relies on estimating the common-mode voltage of the input signal by a simple resistor network and using that voltage as the common reference voltage for the coupled single-ended filters. This method is particularly interesting for high-pass filters because it provides a bias path for the op-amp inputs, but it can also be applied to other filter types, such as band-pass filters. Both design methods proposed yield the desired frequency response, stable response to common-mode voltages, and high CMRR. There is no tradeoff between stability and CMRR, which happens in circuits that rely on resistor or capacitor T-networks [2], [3].
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Oscar Casas (S’93–M’99) received the Ingeniero de Telecomunicación and Doctor Ingeniero de Telecomunicación degrees from the Universitat Politècnica de Catalunya, Barcelona, Spain, in 1994 and 1998, respectively. He is currently an Associate Professor of electronic engineering with the Instrumentation, Sensors, and Interfaces Group, Technical School of Castelldefels, Technical University of Catalonia, Barcelona.Where he teaches courses in several areas of electronic instrumentation. His research includes sensor interfaces, autonomous sensors, electronic instrumentation, noninvasive physiological measurements, and sensors based on electrical-impedance measurements.
R EFERENCES [1] R. Pallàs-Areny and J. G. Webster, Analog Signal Processing. New York: Wiley, 1999, ch. 2. [2] O. Casas and R. Pallàs-Areny, “Basics of analog differential filters,” IEEE Trans. Instrum. Meas., vol. 45, no. 1, pp. 275–279, Feb. 1996. [3] M. Gasulla, O. Casas, and R. Pallàs-Areny, “On the common mode response of fully differential circuits,” in Proc. 17th IEEE IMTC, 2000, vol. 2, pp. 1045–1049. [4] J. M. da Cunha, “A compact and flexible signal conditioning system for data acquisition,” Hewlett Packard J., vol. 45, no. 5, pp. 9–15, Oct. 1994. [5] S. Franco, Design With Operational Amplifiers and Analog Integrated Circuits. New York: McGraw-Hill, 2002. [6] J. Karki, “Fully differential amplifier,” Texas Instruments, Dallas, TX, Application Rep. SLOA054D, 2002. [7] E. M. Spinelli, R. Pallàs-Areny, and M. A. Mayosky, “AC-coupled frontend for biopotential measurements,” IEEE Trans. Biomed. Eng., vol. 50, no. 3, pp. 391–395, Mar. 2003. [8] M. A. Al-Alaoui, “A differential integrator with a built-in high-frequency compensation,” IEEE Trans. Circuits Syst. I, vol. 45, no. 5, pp. 517–522, May 1998. [9] H. Golnabi and A. Ashrafi, “Producing 180◦ out-of-phase signals from a sinusoidal waveform input,” IEEE Trans. Instrum. Meas., vol. 45, no. 1, pp. 312–314, Feb. 1996. [10] R. Casas, O. Casas, and V. Ferrari, “Single-ended input to differential output circuits. A comparative analysis,” in Proc. 23th IEEE IMTC, 2006, pp. 548–551. [11] R. Pallás-Areny and J. G. Webster, “Common mode rejection ratio for cascaded differential amplifier stages,” IEEE Trans. Instrum. Meas., vol. 40, no. 4, pp. 677–681, Aug. 1991. [12] R. Pallás-Areny and J. G. Webster, “Common mode rejection ratio in differential amplifiers,” IEEE Trans. Instrum. Meas., vol. 40, no. 4, pp. 669–676, Aug. 1991.
Marco Massarotto received the degree in electronic engineering and the Ph.D. degree in electronic instrumentation from the University of Brescia, Brescia, Italy, in 2003 and 2007, respectively. Within the electronic instrumentation and measurements fields, his main researches regard differential circuits (voltage filters and charge amplifiers) and the characterization of accelerometers.
Vittorio Ferrari (M’04) was born in Milan, Italy, in 1962. He received the Laurea degree in physics (cum laude) from the University of Milan in 1988 and the Research Doctorate degree in electronic instrumentation from the University of Brescia, Brescia, Italy, in 1993. Since 2006, he has been a Full Professor of electronics with the Faculty of Engineering, University of Brescia. His research activity is in the field of sensors and the related signal-conditioning electronics. Topics of interest are acoustic-wave piezoelectric sensors, microresonant sensors and microelectromechanical systems, autonomous sensors and power scavenging, oscillators for resonant sensors, and frequency-output interface circuits. He is involved with national and international research programs and with projects in cooperation with industries.
Ramon Pallàs-Areny (M’81–SM’88–F’98) received the Ingeniero Industrial and Doctor Ingeniero Industrial degrees from the Technical University of Catalonia (UPC), Barcelona, Spain, in 1975 and 1982, respectively. From 1989 to 1990, he was a Visiting Fulbright Scholar, and from 1997 to 1998, he was an Honorary Fellow with the University of Wisconsin, Madison. In 2001, he was nominated as Professor Honoris Causa by the Faculty of Electrical Engineering, University of Cluj-Napoca, Romania. He is currently a Professor of electronic engineering with UPC and teaches courses in electronic instrumentation. His research includes instrumentation methods and sensors based on electrical-impedance measurements, autonomous sensors, sensor interfaces, noninvasive physiological measurements, and electromagnetic compatibility in electronic systems. He is the author of six books, the leading author of five books, and coauthor of two books on instrumentation in Spanish and Catalan. He is also the coauthor (with J. G. Webster) of Sensors and Signal Conditioning, 2nd ed. (New York: Wiley, 2001) and Analog Signal Processing (New York: Wiley, 1999). Dr. Pallàs-Areny was the recipient, with J. G. Webster, of the 1991 Andrew R. Chi Prize Paper Award from the IEEE Instrumentation and Measurement Society. In 2000, he was the recipient of the Award for Quality in Teaching granted by the Board of Trustees of UPC and, in 2002, the Narcís Monturiol Medal from the Autonomous Government of Catalonia.
