IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 52, NO. 1, JANUARY 2005 49 On the Measurement of Common-Mode Rejection Ratio Jian Zhou, Member, IEEE, and Jin Liu, Member, IEEE Abstract In this brief, several commonly used measurement configurations for common-mode rejection ratio (CMRR) are analyzed and compared with the definition; their advantages and limitations are also discussed. Finally, an improved measurement setup is proposed to characterize CMRR more accurately according to its definition over wider frequency range. Index Terms Amplifiers, common-mode rejection ratio (CMRR), frequency-domain analysis, measurement. I. INTRODUCTION COMMON-MODE rejection ratio (CMRR) is one of the most important performance parameters of high-precision operational amplifiers (opamp) and is defined as the differential voltage gain,, divided by the common-mode voltage gain Alternatively, CMRR can also be viewed as the change in input offset voltage that results from a unit change in common-mode input voltage, as shown [1] [3] where is the input common-mode change and is the input offset voltage change. Intuitively, in a real circuit, a common-mode perturbation at the input may produce a differential component at the output due to the asymmetries in the circuit and the finite output impedance of the tail current source. However, in a feedback system, the differential component at the output is fed back to the input to counteract the effect, so as to make the output constant. The CMRR is a measure of the ability of the circuit to keep the output constant with common-mode perturbation at the inputs. This brief concentrates on the measurement techniques of CMRR. Often, manufacturers data sheets only give dc CMRR value and CMRR curve over frequency while no measurement methods are given. However, as will be shown in Section II, the measured results from various setups actually differ from each Manuscript received February 3, 2004; revised June 8, 2004. This paper was recommended by Associate Editor A. G. Andreou. J. Zhou is with Texas Instruments, Inc., Richardson, TX 75243 USA (e-mail: jian@ti.com). J. Liu is with the Erik Jonsson School of Engineering and Computer Science, Department of Electrical Engineering, University of Texas at Dallas, Richardson, TX 75083-0688 USA (e-mail: jinliu@utdallas.edu). Digital Object Identifier 10.1109/TCSII.2004.838332 (1) (2) Fig. 1. CMRR measurement setup I. other. The limitations of the commonly used CMRR measurement methods are analyzed and discussed. Compared with the CMRR definition, some give smaller CMRR value and some give narrower bandwidth. In Section III, an improved measurement technique is proposed. It is shown that with some post processing, this technique gives more accurate CMRR value over a wider frequency range. II. ANALYSIS OF MEASUREMENT METHODS FOR CMRR A. Setup I Direct Measurement by Definition Fig. 1 shows the configuration that measures CMRR directly according to the definition and it is an intuitive method for simulation. The top configuration measures the differential voltage gain,, while the bottom configuration measures the common-mode voltage gain,. Resistor and capacitor values are chosen to form a low pass filter which provides dc biasing while blocks ac feedback. CMRR can be obtained by dividing by according to the definition. However, measuring high differential voltage gain is not a simple task and is usually done by extrapolating closed-loop gain to dc gain to obtain the full gain curve. In addition, the nature of two measurements to obtain CMRR makes this technique not practical for measurement. B. Setup II Matched Sources Measurement Setup A commonly used method to simulate the CMRR [4] employs matched sources as shown in Fig. 2. Two identical voltage sources designated as are placed in series with both opamp 1057-7130/$20.00 2005 IEEE
50 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 52, NO. 1, JANUARY 2005 Fig. 2. CMRR measurement setup II. (a) Configuration for direct simulation of CMRR. (b) Small-signal model. Fig. 4. Comparison of CMRR measurement setup I and setup III. Fig. 3. CMRR measurement setup III. inputs where the opamp is connected in the unity-gain configuration. The small-signal model of this circuit is shown in Fig. 2(b). It can be shown that (3) In the denominator, the unity and the term are generally smaller than in the frequency range of interest, so the approximation is valid and the output is directly related to CMRR. This technique is very convenient for simulation; however it may not be practical for ac measurement. The two sources applied at both inputs have to be equal and in phase, while in measurement it is not easily guaranteed. C. Setup III Power Supply Measurement Setup Another commonly used configuration [5], [6] is shown in Fig. 3. A common-mode perturbation is applied to both the positive and negative supplies. Intuitively, the bias voltage applied at the positive input serves as the baseline common-mode voltage, a perturbation applied at both the positive supply and negative supply acts as the common-mode perturbation to both the positive input and the negative input. CMRR is evaluated at the output as However, if we compare the simulation result of CMRR using measurement setup III with that using setup I (direct measurement by definition ), the two results are not quite the same as shown in Fig. 4. The line with square ticks is the simulation result by dividing differential voltage gain by common- (4) Fig. 5. (a) CMRR measurement setup III. (b) Equivalent circuit. (c) Smallsignal model. (d) Derivation of CMRR. mode voltage gain as the measurement setup I in Fig. 1. The line with the triangle ticks is the simulation result from measurement setup III in Fig. 3. The other two curves are differential voltage gain and common-mode voltage gain. We can see that measurement setup I gives wider bandwidth and higher value than measurement setup III. The simulation results are carried out using Spice with transistor level circuits of the amplifier. The amplifier used is a Texas Instruments catalog part of general purpose operational amplifiers. To find why simulation setup III gives narrower bandwidth and lower value, let us look at the small-signal model of the setup III shown in Fig. 5. From the derivation, we obtain The unity term in the denominator will dominate when common-mode gain drops close to or even below unity, thus, contributes to the reduction of the bandwidth. If common-mode voltage gain is less than unity at dc, this technique give lower (5)
ZHOU AND LIU: ON THE MEASUREMENT OF CMRR 51 Fig. 7. CMRR measurement setup IV. Fig. 6. DC CMRR measurement setup. dc CMRR than definition. Intuitively, since the reference point is moved to above the ground, the output changes to the reference point which is different from the definition to keep the output constant, that is why the unity appears in the denominator. This also explains why measurement setup III gives lower value and narrower bandwidth. D. DC CMRR Measurement Setup There is a commonly used dc CMRR measurement setup [7], [8] as shown in Fig. 6. The device under test (DUT) is configured in a servo loop. The servo amplifier should have high gain and wider bandwidth than that of the DUT to keep the loop stable. is a fixed bias voltage to set the DUT output voltage, preferably half of the supply voltage. Since the differential voltage appears between the positive and the negative inputs of the DUT is much smaller compared to the change of, we can assume the common-mode change at the DUT input is and derive that where, is the gain from the servo amplifier input to the servo amplifier output including the feedback capacitor and resistor. This gain is equal to opamp open-loop gain and very high at dc and drops to unity at high frequency. We know that the second term on the left-hand side is much larger than the right-hand side, so Hence We can see that the servo loop forces the DUT output at at dc which meets the CMRR definition to keep the output constant. Hence CMRR value from this test is very close to the theoretical one. (6) (7) (8) However, when the frequency increases, the gain of the servo loop quickly drops to unity. The -db frequency is determined by, given the values as shown, the frequency is only about 1.6 KHz. At frequency higher than 1.6 KHz, the right-hand side term in (6) is significant and can no longer be neglected. Then, (6) becomes Since (9) (10) (11) Because the unity term in the denominator, we will see the same effect of narrow bandwidth as in the simulation setup III using this configuration. E. Setup IV Matched Resistor Measurement Setup Another commonly used CMRR test fixture [7] is shown in Fig. 7. The test circuit is basically a difference-amplifier configuration with the two inputs tied together. and are two matched resistors and and are two matched resistors as well. For simplicity, is set to ground. There are two major problems with this technique for the measurement of opamp CMRR. The first one is that the resistors must be known precisely and carefully matched. A CMRR value of 100 db would require resistor matching roughly to 0.001%, an impractical value to achieve in practice. With four equal resistors having matching of 0.01%, the highest CMRR achievable is 86 db, the CMRR curve looks like truncated at high CMRR as shown below. A remedy to this problem is to measure the dc CMRR value from the dc CMRR measurement setup in the previous section, and then to extrapolate the measured CMRR curve to obtain the full CMRR curve as shown in Fig. 8. The second problem with this technique is the omission of output impedance of the operational amplifier. Resistors and output impedance consist of a forward path, take this forward path into account, there are three paths
52 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 52, NO. 1, JANUARY 2005 Fig. 9. CMRR measurement setup V. Fig. 8. CMRR results using measurement setup IV. from input to output: differential gain, common-mode gain, and feedforward, respectively (12) (13) Assume that,we have Fig. 10. Comparison of CMRR measurement setup I (definition), setup IV (without buffer) and setup V (proposed method with buffer). Hence, Let us assume that and let to simplify the expression. (14) (15) (16) We see a similar expression again. If, (16) becomes (17) which is expected to have similar bandwidth reduction as that in measurement setup III. III. IMPROVED CMRR MEASUREMENT SETUP WITH UNITY GAIN BUFFER We explained that the nonideality on setup IV is caused by the finite output impedance of the DUT amplifier. If we can minimize the outputimpedance, we canminimize this nonideality. Inserting a unity gain buffer will serve this purpose as shown in Fig. 9. The negative feedback in the unit gain buffer reduces the impedance looking into node by times, where is the differential voltage gain of the amplifier, A2. As a result, the output impedance is greatly reduced. If amplifier A2 has wide enough bandwidth over the frequency range we are interested, (16) becomes (18) Fig. 10 shows the comparison between different measurement setups. The line with square ticks is the CMRR obtained by dividing common-mode voltage gain from differential voltage gain. The line with triangle ticks is CMRR result from the measurement setup V with output buffer as shown in Fig. 9. The line with rectangle ticks is the CMRR result from the measurement
ZHOU AND LIU: ON THE MEASUREMENT OF CMRR 53 setup IV without output buffer as shown in Fig. 7. We can see that there is significant discrepancy between measurement setup IV and V; the CMRR obtained by measurement setup V is much closer to measurement setup I. Resistor matching is still required in this method, the extrapolating technique discussed above can be used to correct any resistor mismatch and obtain the full curve. IV. CONCLUSION We have analyzed several CMRR measurement setups and their results and we have shown the discrepancy between the commonly used setups for measuring CMRR and the definition of CMRR. An improved setup was proposed to characterize CMRR very close to the definition, and over wider bandwidth. ACKNOWLEDGMENT The authors wish to thank F. Tsay and G. Romas for their discussion and valuable inputs to the brief. REFERENCES [1] P. Gray and R. Meyer, Recent advance in monolithic operational amplifier design, IEEE Trans. Circuits Syst., Fundam. Theory Appl., vol. CAS-21, no. 3, pp. 317 327, May 1974. [2] B. Razavi, Nonlinearity and mismatch, in Design of Analog CMOS Integrated Circuits. New York: McGraw-Hill, 2001, ch. 13, p. 478. [3] P. Gray and R. Meyer, Operational amplifier, in Analysis and Design of Analog Integrated Circuits, 3rd ed. New York: Wiley, 1993, ch. 16, p. 422. [4] P. E. Allen and D. R. Holberg, Unbuffered CMOS op amps, in CMOS Analog Circuit Design. Oxford, U.K.: Oxford Univ. Press, 1987, ch. 8, p. 430. [5] M. E. Brinson and D. J. Faulkner, New approach to measurement of operational amplifier common-mode rejection ratio in the frequency domain, in Proc. IEE, Circuits Devices Syst., vol. 142, 1995, pp. 247 253. [6] Recommended test procedures for operational amplifier, Intersil, Milpitas, CA, Appl. Note 551, Nov. 1996. [Online.] http://www.intersil.com/data/an/an551.pdf. [7] M. Burns and G. Roberts, An Introduction to Mixed-Signal Test and Measurement. Oxford, U.K.: Oxford Univ. Press, 2001. [8] J. Bryant and W. Kester, Operational amplifiers, in Linear Design Seminar Notes. Norwood, MA: Analog Devices, 1995.