THE increased complexity of analog and mixed-signal IC s

134 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 34, NO. 2, FEBRUARY 1999 An Integrated Low-Voltage Class AB CMOS OTA Ramesh Harjani, Member, IEEE, Randy Heineke, Member, IEEE, and Feng Wang, Member, IEEE Abstract We present a new circuit topology for a low-voltage class AB amplifier. The circuit shows superior current efficiency in the use of the supply current to charge and discharge the output load. It uses negative feedback rather than component matching to optimize current efficiency and performance, resulting in a current boost ratio exactly equal to one. Measurement results for an example circuit fabricated in a 2-m CMOS process are given. The circuit uses a quiescent supply current of 0.2 A and is able to settle to a 1% error in 1.1 ms for a 0.4-V input step and a load capacitance of 35 pf. The circuit design is straightforward and modular, and the core circuit can be used to replace the differential pair of other op-amp topologies. Index Terms Class AB, CMOS operational transconductance amplifier (OTA), current utilization, low voltage. Fig. 1. Circuit model used to illustrate current utilization. I. INTRODUCTION THE increased complexity of analog and mixed-signal IC s makes it crucial to optimize the power-to-performance ratio of such circuits. Additionally, there is a shift toward lower voltage operation, making such optimization even more critical. To achieve both these goals, new circuit topologies and building blocks need to be explored. The operational amplifier (op-amp) is one of the most important and widely used building blocks [1]. Class A op-amps have poor largesignal behavior since their tail currents are constant, resulting in a slew-limited transient response. An alternative solution is to use class AB op-amps. They have a low and well-controlled quiescent current, which is automatically boosted when a large differential voltage is applied. There are basically two categories of class AB op-amps. In the first category, the tail current of the input differential transconductance transistors is boosted for large signals [2] [4], [7]. This results in increased current for the entire op-amp during large-signal operation. For the second category, only the current in the output stage is boosted for large signals [5], [6]. In [5] and [6], the output current is boosted by measuring the differential input voltage. A large differential input voltage results in a large boosted output current. This paper concentrates on input tail-current boosted class AB op-amps. Among these, one of the earliest designs uses source-coupled NMOS and PMOS transistors [3]. However, this circuit is not well suited for low-voltage operation because of the stacked NMOS and PMOS threshold voltages. The rest of the circuits in this category utilize some form of positive feedback to boost the tail current when required. Among them, the adaptive biasing amplifier [4] is based on Manuscript received September 24, 1997; revised June 29, 1998. The authors are with the Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455 USA. Publisher Item Identifier S 0018-9200(99)00975-0. a class A op-amp and uses additional circuitry to obtain the current difference through the input transistor pair. This current difference is then fed back as additional tail current. The feedback current increases with increased input voltage. The asymmetrical current boost circuit in [7] uses a current mirror to return a fraction of the current through the input pair as additional tail current. Therefore, when a large differential voltage is applied, the total tail current increases. The rest of this paper is organized as follows. We first introduce the concept of current utilization for the various op-amp designs. This provides the motivation for the new circuit design. This section on current utilization is followed by the principle of operation and circuit details for our design. We first describe the circuit operation in strong inversion and then extend it to weak inversion. We then present some experimental results and some final conclusions. II. CURRENT UTILIZATION An important consideration of power management lies in the portion of supply current that is delivered to the load, i.e., the current utilization. The following paragraphs examine the current utilization for the proposed circuit and the other established class AB topologies with a single stage, current boosted differential input pair, and continuous operation. Current utilization is limited by three common circuit features. To illustrate the current utilization factor, we use the circuit shown in Fig. 1. In this figure, is the output current mirror ratio. This is the ratio of the current in the output load compared to the current difference in the input differential pair. Likewise, is the current boost ratio, particular to this category of class AB amplifiers. This is the ratio of the change in tail current to the change in the differential pair output current. The other circuit feature, not specifically marked in this figure, is 0018 9200/99$10.00 1999 IEEE

HARJANI et al.: LOW-VOLTAGE CLASS AB AMPLIFIER 135 Fig. 2. Circuit from [4]. Fig. 4. Circuit used to analyze current utilization for step inputs. Fig. 3. Circuit from [7]. the minimum number of copies of the differential pair current required for a particular topology. In general, for all the other single-stage class AB amplifiers, is also the positive feedback current ratio that is dependent on device matching. Our circuit, on the other hand, uses negative feedback to guarantee is always at its optimal value of unity. Other values of lead to either wasted current or loss of performance. Values of larger than one are wasteful and increase the current in both sides of the differential pair, for large signals. The output load only sees the difference between the currents at the two sides of the differential pair. Therefore, any increase in both sides is wasteful and only serves to increase power consumption. If is less than or equal to one, the current for one side of the differential pair will increase to a point where it overwhelms the negligible current in the opposite side. For the different choices of, the portion of wasted tail current is given by (1) where is the tail current and is the amount of tail current that does not contribute to load current. Values of less than one restrict the ultimate performance by limiting the peak current capability of the differential pair, and as such, the peak current capability of the overall opamp. For values of greater than one, the peak current is unbounded. The value for the peak current can be expressed as follows: where is the quiescent tail current. We compare the adaptive bias amplifier [4] (shown in Fig. 2), the asymmetrical current boost amplifier [7] (shown for for for for (1) (2) Fig. 5. Topology comparison for B =3;=0:5; =2: in Fig. 3), and our new topology (shown in Fig. 10). The ratio for all these topologies commonly ranges between one and ten [8], with three being a useful value for comparison. The definition of takes into account the current in the differential pair and each of its copies that does not supply the load. The values of are three for [4], two for [7], and one for the new circuit. The current utilization limit when is calculated by the ratio of the load current to the supply current. The load current is times the differential pair current, while the supply current is times the differential pair current Maximum efficiency (3) The graphs shown in Figs. 5 8 illustrate the impact of these three variables on current utilization for a range of differential input voltages. One of the uses of these relationships is to evaluate the current utilization for a transient response. For example, we use the circuit in Fig. 4 to analyze current utilization for a step input. The input voltage takes a step of size However, the output voltage cannot change instantaneously. Hence, the differential voltage jumps to, which provides the stimulus required by the amplifier to supply current into the load. The output voltage increases rapidly and settles to the input voltage, with the result that the differential input voltage falls from to zero. Correspondingly, in Figs. 5 8, during settling the current utilization follows a trajectory from the right to the left along the appropriate curve.

136 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 34, NO. 2, FEBRUARY 1999 Fig. 9. Circuit schematic for the self-biased input transistor. Fig. 6. Topology comparison for B =3; =0:95; 1:05: Fig. 7. Topology comparison for B =1; =0:95; 1:05: Fig. 8. Topology comparison for B =10; =0:95; 1:05: Fig. 5 shows a comparison of the three topologies with typical values for and quiescent current. In this figure, we have attempted to focus on small differential input voltages to expose the details around the origin. For each input step, even though the proportion of time spent at large input differentials is small, the total power consumption is usually proportional to the large-current efficiency toward the right of the figure. This is because the total current is quadratically (for strong inversion) or exponentially (for weak inversion) related to the input voltage and dominates for reasonable inputs. From this figure, it is easy to see that the current utilization for the new circuit approaches the 75% limit. However, the current efficiency for the Callewaert et al. circuit [7] increases more rapidly at first but limits at a lower value. The higher current efficiency for small inputs results from the use of asymmetrical differential pairs. However, current utilization is limited due to the second current copy. Its maximum efficiency approaches the 60% limit when is one-half. The reduction of current efficiency, when is two, is clearly seen for large inputs. Degrauwe et al. s circuit [4] is less current efficient because of the third current copy that is required for this circuit to operate. Again, a reduction of current efficiency results for values of greater than one. Fig. 6 shows the same comparison with values of that are representative of process variation. This graph is more representative of real-world design choices for the various parameters. It is worth noting that current utilization is not seriously degraded for values of that are slightly greater than one even though stability is. Figs. 7 and 8 show that large values of increase the current utilization and minimize the differences of current utilization for the different topologies, while small values of reduce and exaggerate the differences of current utilization, further suggesting that the value of should be maximized. However, increased results in lowered phase margin and therefore cannot be made too large [8], [9]. It is clear from the graphs shown above that the new circuit has a better current-utilization factor for larger inputs in comparison to previous designs. Since the total power consumption is dominated by the peak current, the current efficiency for large differential inputs is most critical. We now proceed to provide design details for the new circuit. III. NEW OPERATIONAL TRANSCONDUCTANCE AMPLIFIER CIRCUIT DESIGN Before we explain the basic operation of our class AB amplifier [2], let us first understand the operation of the proposed self-biasing input transistor shown in Fig. 9. A. Self-Biased Input Transistor Transistor is the input transistor, and transistor is the auxiliary input transistor. Operational transconductance amplifier and transistor form a negative feedback loop. The negative feedback mechanism can be understood as follows. Suppose the source voltage of transistor increases

HARJANI et al.: LOW-VOLTAGE CLASS AB AMPLIFIER 137 Fig. 10. Overall circuit for class AB op-amp. slightly; consequently, the output voltage of will also increase. This increment will force voltage to go down. This mechanism enables the source voltage of transistor to be virtually equal to the source voltage of transistor As and now have the same gate and source voltages, the current density will also be the same for these two transistors if we assume ideal MOS behavior and neglect channel-length modulation. By correctly sizing the device dimensions of transistors and, the choice of, the quiescent current flowing through transistor, can be well controlled. The output voltage of operational transconductance amplifier (OTA) automatically adjusts to bias transistor so as to generate the requisite quiescent current. In [7], a similar self-biasing differential input pair is proposed. There are several differences between our proposed technique and the one in [7]. First, although both techniques include a feedback loop, our technique uses a negative feedback loop while [7] uses a positive feedback loop. Second, we use a single selfbiasing input transistor as a basic building block, while [7] uses the self-biasing differential input pair. Two of our building blocks can be combined into a differential pair, as explained in the next paragraph. In [7], due to the fact that the role of the two input transistors in the self-biasing differential input pair is not interchangeable, an additional self-biasing differential input pair with opposite input polarity is required to form a complete op-amp. As a consequence, a fairly complex topology results. Additionally, if the current mirror is perfectly matched and the current feedback ratio is equal to one, then the output current is proportional to the square of the input voltage difference for strong inversion operation. Unfortunately, if the current feedback ratio is greater than one, the circuit becomes unstable. Therefore, to accommodate finite device mismatch, the current ratio has to be less than one to avoid instability. Then, the output current is no longer exactly proportional to the square of the input voltage difference. B. Principle of Operation The basic operation for the proposed class AB amplifier can now be illustrated with the help of Fig. 10. The circuit is based on the principle of the self-biased input transistor. Two such transistors are combined in a common source configuration. Except for the input stage and tail current, the new circuit resembles the symmetrical OTA. Note that the proposed input pair can be extended to other circuit topologies for OTA s without much effort. The key to understanding the circuit operation is to recognize the negative feedback mechanism that ensures that the common source voltage of and is always forced to track the smaller of the two voltages and Without any loss of generality, let us assume that the gate voltage is larger than the gate voltage Therefore, voltage is larger than Effectively, the output voltage of op-amp is pulled down, which turns off transistor Due to the negative feedback, the drain voltage of transistor is equal to the voltage In other words, the voltage at the common source of and is constant with respect to the gate voltage of and is only determined by the magnitude of the device size of transistor and the constant bias current A SPICE simulation of this condition is shown in Fig. 11. This simulation shows the voltages and Voltage is kept constant, and voltage is altered. In the figure, we note that the voltage at first follows, and then, when is greater than, it follows Using the previous conditions, the current through be written as shown in (4) Further, using the relationship between and i.e., the expression for the current can be rewritten as can where We neglect the back-gate effect and assume is equal to The above equation (4) (5) (6)

138 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 34, NO. 2, FEBRUARY 1999 Fig. 11. Fig. 12. Common source voltage versus positive input voltage. Output current versus differential input voltage. In [3], a source-coupled NMOS and PMOS transistor pair is used to generate a current that is proportional to the square of the differential input voltage. However, this circuit suffers from limited common-mode input range and is not suitable for low-voltage operation. The minimum supply voltage for this circuit is equal to [10]. For our proposed circuit topology, the minimum supply voltage is equal to Given a typical value of of 0.2 V (edge of strong inversion) and of 0.7 V, the former expression yields 2 V and the latter expression yields 1.3 V. The common input voltage range for our circuit is similar to that of a conventional class A op-amp. It is clear that the proposed topology is suitable for low-voltage operation. The maximum current through the input transistors can also be estimated. Again, we assume that the gate voltage of transistor is larger than that of transistor The source voltage is, which is determined by both the gate voltage and the bias current To ensure that transistor operates in saturation, the following relationship needs to hold: (8) (9) where is the drain voltage of transistor, is the threshold voltage for PMOS transistors and has a negative value, and Therefore, the maximum achievable input voltage without entering triode operation region and the maximum current are as follows: (10) (11) shows that is proportional to the square of the differential input voltage. A similar expression can be found in [1] and [3]. The maximum current, however, is limited by the supply voltage and the maximum current that transistor can sink. The current through transistor is the bias current multiplied by the ratio of the device sizes of and By interchanging and in (6), we can derive an expression for the current through transistor when is larger than A SPICE simulation of the output current versus differential input voltage is shown in Fig. 12. For this design, the quiescent current was set to 0.4 A, and the device ratio of transistors was set to ten. Note, however, that as the differential input voltage increases to 1 V, the output current increases to approximately 800 A. The quiescent current of the input stage is very well controlled. For a zero differential input voltage, is equal to and the common source voltage of the input pair and is equal to Therefore, the quiescent current through the transistor is given by (7). The quiescent current for the complete circuit can be kept small by selecting a small value for (7) By examining (11), we can see that not surprisingly, the largest current results when the difference between is maximum. However, voltage cannot be made too small. Otherwise, transistor will be forced into triode operation. For a given maximum current the minimum will be equal to By solving the above two equations simultaneously and keeping in mind that and are functions of the expression for the maximum current can now be written as shown in (12) where (12) (13) The maximum current can be made large if device size ratios of transistors and are made large with respect to Unfortunately, the gate capacitances associated with these transistors increase as well with device size. Therefore, a design tradeoff needs to be made because these capacitances determine the second pole of the overall circuit and also affect how rapidly the gate of transistor can be charged and discharged.

HARJANI et al.: LOW-VOLTAGE CLASS AB AMPLIFIER 139 Fig. 13. Circuit schematic for negative feedback op-amp. 1) Extension to Weak Inversion: The proposed class AB op-amp design can easily be extended to operate in weak inversion. Assuming that the input transistors operate with larger than a few, we can rewrite (6) as shown in (14) Fig. 14. Microphotograph of the chip. (14) where is the weak inversion slope factor and is the thermal voltage. When only one gate voltage, i.e., is swept, the current will be kept constant. Therefore, the current will increase exponentially with the differential input voltage as opposed to quadratically when the input transistors operate in strong inversion. For larger input voltages, the transistors will go into strong inversion and follow the quadratic behavior described in (4). If we denote as the differential input voltage and as the corresponding peak current, then we have the following relationship regarding peak current versus input step size: (15) The above expression can be experimentally verified by measuring the peak currents for different input step sizes. Note that in practice, the actually measured current peak ratio is likely to be somewhat smaller than predicted by (15) because when one gate voltage takes a step jump, the common source voltage is not really fixed. It will tend to follow the gate voltage, which in turn reduces the effective step size. A simple five-transistor OTA is used for the negative feedback op-amp and is shown in Fig. 13. Because the input voltage of the feedback op-amp is close to the negative rail, PMOS input transistors are used. More complex topologies can be utilized for these op-amps if desired; however, the simple structure shown was sufficient to meet our design specifications at low currents. 2) Negative Feedback Loop: As mentioned before, the common source voltage of transistor and tends to track the smaller value of and This tracking function is provided by the negative feedback mechanism. The behavior of responding to any change of voltage at the negative node of op-amp will directly affect the performance of the proposed class AB op-amp. The closed-loop behavior for the loop can be written as shown in (16) (16) where and are the transconductances of the input transistor inside input transistor and transistor respectively. and are the total gate and drain capacitances of and The natural frequency and loop is given by and For a critically damped transient response, value of needs to equal 0.5. This determines the maximum, which is IV. EXPERIMENTAL RESULTS The new class AB op-amp shown in Fig. 10 was fabricated using a MOSIS 2- m N-well CMOS technology. The chip microphotograph is shown in Fig. 14. Unless otherwise specified, the test results were measured with the op-amp biased with a supply of 2.4 V and a quiescent tail current of 56 na. Fig. 15 shows the dc transfer characteristic of the class AB op-amp when operated in the unity-gain feedback mode, as well as the offset voltage between the input and output voltages. The op-amp has a linear output range from 0.5 to 2.35 V. The measured offset voltage in the linear region is less than 20 mv. Due to the finite gain of the OTA, the expected gain-related offset voltage is where is the dc gain of the op-amp. Since the op-amp only has a single gain stage without any cascode transistor, the dc gain is expected to be around 40 db, which is confirmed in Fig. 17. After performing a straight line fit, the random component of the offset was found to be approximately 5 mv for a dc input of 0 V.

140 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 34, NO. 2, FEBRUARY 1999 Fig. 15. DC transfer characteristics of the op-amp in unity-gain mode. Fig. 18. Transient response of the amplifier to a step input. Fig. 16. voltage. Measured input common-mode range as a function of supply Fig. 19. Transient behavior of current supply for a step input (0.4 V). Fig. 20. Transient behavior of current supply for a step input (0.3 V). Fig. 17. Open-loop ac transfer characteristics of the op-amp. Fig. 16 shows the measured input common-mode range as a function of the supply voltage. For this experiment, the supply voltage was swept from 1 to 2.5 V. For each value of supply voltage, the input common-mode range was measured using the unity-gain configuration discussed above. We note that the input common-mode range is a linear function of the supply voltage. A supply voltage of 1 V results in an input commonmode range of approximately 0.2 V. This value compares well with the 1.3-V minimum supply voltage calculated in Section III-B for strong inversion operation, verifying that our new design behaves like an ordinary differential pair as far as common-mode range is concerned. Our lower measured

HARJANI et al.: LOW-VOLTAGE CLASS AB AMPLIFIER 141 TABLE I OP-AMP MEASUREMENT RESULTS values are a result of operation in weak inversion and lower threshold voltages for our process. Fig. 17 shows the ac characteristic of the class AB op-amp. The measured dc gain is around 42 db. The gain bandwidth is around 100 khz, and the phase margin is greater than 75 for a 34-pF load. If instead a cascoded output stage is used, the dc gain is expected to increase approximately 100-fold. No other parameters other than the dc gain, the gain-related offset component, and the output voltage range are expected to change if the output is cascoded. The small signal characteristics of the proposed class AB op-amp are comparable to those obtained with conventional amplifiers. The transient behavior for the op-amp is shown in Fig. 18. The op-amp is configured in unity-gain mode with a capacitive load of 47 pf and a step applied to the input. Please note that the midpoint levels for traces 1 and 2 have been separated for clarity. The supply current is monitored at the same time and is shown in Fig. 19 with a compressed time scale to more clearly show the consistency of the current peaks for transients in the same direction. From the figure, we can see that the supply current of the op-amp (large spike) increases substantially when a large input step is encountered, even with a low quiescent current, as might be expected for class AB amplifiers. The peak current versus quiescent current ratio is estimated to be around 100 for a 0.4-V step. Fig. 20 shows the supply current for a step input of 0.3 V. In this case, the current peak is approximately one-quarter the value for the 0.4- V step input. An intuitive explanation for this operation can be provided with the help of (15) since the input transistors initially operate in weak inversion. Using (15), the numerical value for is approximately 0.065 V, i.e., the effective difference between the step inputs is 0.065 V. However, the actual difference between the step inputs is 0.1 V (0.4 0.3 V). The discrepancy between the expected value and measured value is due to two factors. One is the reduced effective input step, as mentioned before. The other is that one of the input transistors moves out of weak inversion into strong inversion when the current increases dramatically, in which case the relationship between the input steps and the output current is quadratic instead of exponential. A summary of the measured results is shown in Table I. V. CONCLUSIONS We have presented a new class AB op-amp circuit topology suitable for low-voltage operation. The circuit shows superior current efficiency in the use of the supply current to charge and discharge the output load. The fabricated example circuit of this class AB op-amp in a 2- CMOS process demonstrates an approximately 500-fold increase in the bias current for largesignal operation. Due to its superior large-signal performance, the core circuit can also serve as an input stage of a buffer amplifier. The circuit uses negative feedback to guarantee an the current boost ratio, of one for most efficient current usage. The design is straightforward and modular. The core circuit (shown in bold lines in Fig. 10) can also be used to replace the differential pair of a large number of op-amp topologies. The resulting circuits would all operate as class AB amplifiers. REFERENCES [1] F. Wang and R. Harjani, Optimal design of opamps for oversampled converters, in Proc. IEEE Custom Integrated Circuit Conf., 1996, pp. 337 340. [2] F. Wang, R. Heineke, and R. Harjani, A low voltage class AB CMOS amplifier, in Proc. IEEE Int. Symp. Circuits and Systems, 1996, pp. 392 396. [3] R. Castello and P. R. Gray, A high-performance micropower switched-capacitor filter, IEEE J. Solid-State Circuits, vol. SC-20, pp. 1122 1132, Dec. 1985. [4] M. Degrauwe, J. Rijmenants, E. A. Vittoz, and D. Man, Adaptive biasing CMOS amplifiers, IEEE J. Solid-State Circuits, vol. SC-17, pp. 522 528, June 1982. [5] K. Nagaraj, CMOS amplifiers incorporating a novel slew rate enhancement technique, in Proc. IEEE Custom Integrated Circuit Conf., pp. 11.6.1 11.6.5, 1990. [6] R. Kline, B. J. Hosticka, and H. J. Pfleiderer, A very-high slew rate {CMOS} operational amplifier, IEEE J. Solid-State Circuits, vol. 24, pp. 744 746, June 1989. [7] L. Callewaert and W. Sansen, Class AB CMOS amplifiers with high efficiency, IEEE J. Solid-State Circuits, vol. 25, pp. 684 691, June 1990. [8] M. Degrauwe and W. Sansen, The current efficiency of MOS transconductance amplifiers, IEEE J. Solid-State Circuits, vol. SC-19, pp. 349 359, June 1984. [9] R. Wang and R. Harjani, Partial positive feedback for gain enhancement of low power CMOS OTA s, Analog Integrat. Circuits Signal Process., vol. 8, no. 1, pp. 21 35, July 1995. [10] J. Huijsing, Analog Circuit Design. Norwell, MA: Kluwer Academic, 1993. Ramesh Harjani (S 87 M 89) received the B.Tech. degree from the Birla Institute of Technology and Science, Pilani, India, in 1982, the M.Tech. degree from the Indian Institute of Technology, New Delhi, India, in 1984, and the Ph.D. degree in electrical engineering from Carnegie-Mellon University, Pittsburgh, PA, in 1989. He was with Mentor Graphics Corp., San Jose, CA, and worked on CAD tools for analog synthesis and power electronics. He joined the University of Minnesota, Minneapolis, in 1990 and is currently an Associate Professor in the Department of Electrical Engineering. His research interests include low-power analog design, sensor interface electronics, analog and mixed-signal circuit test, and low-power wireless communications circuits. Dr. Harjani received the National Science Foundation Research Initiation Award in 1991 and a Best Paper Award at the 1987 IEEE/ACM Design Automation Conference. He was an Associate Editor for IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS PART II: ANALOG AND DIGITAL SIGNAL PROCESSING from 1995 to 1997.

142 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 34, NO. 2, FEBRUARY 1999 Randy Heineke (S 96 M 97) was born in Cincinnati, OH, on May 10, 1955. He received the B.Eng. degree in electronics from the University of Cincinnati in 1979. He currently is pursuing the M.S. degree in electrical engineering at the University of Minnesota, Minneapolis. From 1979 to 1994, he was with IBM Corp. His interests are in the design of analog integrated circuits. Feng Wang (M 95) received the B.S. degree in physics from the University of Science and Technology of China, P. R. China, in 1989 and the M.S. and Ph.D. degrees in electrical engineering from the University of Minnesota, Minneapolis. Since 1997, he has been with the Measurement Division of Rosemount, Inc., Eden Prairie, MN, where he is a Senior Design Engineer and is involved in the development of analog/digital mixedsignal IC s for sensor interfaces. He is an Adjunct Professor in the Department of Electrical Engineering at the University of Minnesota, Minneapolis. He is a coauthor of Design of Modulators for Oversampled Converters (Norwell, MA: Kluwer, 1997). His primary research interests are in the area of oversampled converters and communication circuits.