Figure 2 shows a plot of the counting rate as a function of boiling time of the sample to accomplish the reduction of Cr(V1) at several different mole ratios of sodium bisulfite to Cr(V1). It is evident from this plot that the uptake of Cr(V1) is quantitative after five minutes boiling at a mole ratio from 5 0 : l to 1OO:l. At mole ratios below 50:1, the uptake is slower. This is believed to be caused by a relatively slower redox reaction and air oxidation of the sodium bisulfite leading to a slower rate of formation of Cr(II1). It is observed that a t long boiling times, there is a significant uptake of the chromium in Cr(V1) in the absence of sodium bisulfite. This phenomenon has been observed in the case of EDTA by several groups. Belin and Fouchecount (17) have discussed the reduction of Cr(V1) in the presence of EDTA. Their con(17) P. Belin and P. Fouchecount, Comp. R e d . Ser. C. 263, 539 (1966).
clusion is that the reduction of Cr(V1) is favored at low p l l values and high temperatures. At a pH of 1.5 and a teinperature of S0”C with a mole ratio of EDTA to Cr of 30:1, there is total reduction of Ci(V1) to Cr(lI1) in 1.5 hours. If the pH is raised to 3.5 and all other parametcrs are kept constant, the reduction is complete in 2.5 hours. However, at room temperature, the reduction reaction requires several months to reach completion. In general, this result has been found in the case of Chelex 100. Again, the reduction seems to be somewhat faster with Chelex 100; however, it is slow enough to allow for the analysis of Cr(II1) in the presence of Cr(V1) at room temperature. RECEIVED for review May 18, 1971. Accepted September 22, 1971. This work was supported in part by Research Grant GP-24311 from the National Science Foundation.
Analysis of Operational-Amplifier Drift Byrg E. Bonnelycke Chemistry Laboratory I I I , H. C . Orsted Institute, Uniaersity of Copenhagen, Copenhagen, Denmark
THEOPERATIONAL AMPLIFIER is a versatile tool in the hands of the instrumental chemist who understands its shortcomings ( I , 2 ) . Among these shortcomings, the “drift” can be dealt with mathematically-at least in some operational-amplifier circuits-if one knows how to include the manufacturers’ data into the equivalent circuits. Since handbooks and textbooks tell very little about how to analyze circuits with drift, it is worth dwelling on this subject. The Ideal Operational Amplifier. The characteristics of the ideal operational amplifier are indicated in FiguIe 1. The input impedance, R,,, and the open-loop amplification, A,, are infinite at all frequencies and signal levels (i,e., no saturation occurs in the output), and the output impedance, R,, is zero. Especially important from the viewpoint of drift: there is zero output-signal with zero input-signal. “Zero input-signal” of course does not mean that the input terminals are left floating since then pick-up voltages could develop across the high input impedance; it means that the input terminals are shorted, which simulates the zero-signal condition in any external signal source. The Real Operational Amplifier. In the representative interface circuitry for the real operational amplifier on the left in Figure 2, it is clear thal there are several reasons why the output voltage, E,, may differ from zero, even though the input voltage is zero (e.g., grounded). The first cause is power-supply drift. If B+ and B- drift cooperatively in the same direction, the bias point of the output transistor will tend to follow. Thus, when high accuracy is required, it is necessary to invest in a high-quality power supply with negligible drift. The second and more disastrous cause is bias-point drift in the input transistors, Q1and Q2. This is due to either ambient temperature fluctuations around the transistors or differences in heat dissipation due to drive level. This effect is particularly cumbersome because the very high open-loop gain required of a n operational amplifier (104-106) will am(1) John S. Springer, ANAL.CHEhf.,42 @), 22A (1970). (2) Richard G. McKee, ihid., (ll), p 91A.
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ANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
i .
I
1
I
R,, = 00
R, = 0 A0
=
OO
Figure 1. Equivalent circuit af ideal operational amplifier plify not only the desired signals in the input stage but also any unwanted voltage fluctuations of the bias points of the two input transistors. Increasing the open-loop gain, A,, by adding another internal stage of amplification is n o solution since that part of the open-loop output drift which originates in the input stage is subjected to the same increase in gain. In fact, assuming perfect power supplies, the open-loop driftto-gain ratio is a constant system parameter of the operational amplifier and depends on the quality of the input stage. These fluctuations, due primarily to temperature, are properly referred t o as drift (3). The third cause why the output voltage fluctuates is due to internal component-noise, e.g., thermal noise within the input resistors as well as transistor noise. These random voltages, caused by the microscopic granularity or discontinuity of current flow, are properly referred t o as noise-not drift. These noise sources would still operate even if both power supply and temperature stabilization were perfect (4). However, since the low-frequency cf < l KHz) transistor noise (4, 5) is inversely proportional to frequency (l/f noise), the (3) P. J. Beneteau, G. M. Riva, B. Murari, and G. Quinzio, “Drift Compensation in DC Amplifiers,” Solid State Desigiz, May
(1964). (4) Seymour Letzter and Norman Webster, ZEEE Spectrum, 7, 67 (1970). ( 5 ) R. Ralph Benedict, “Electronics for Scientists and Engineers,” Prentice-Hall, New York, N.Y., 1967, p 13.
B+
t
S
I Figure 2. amplifier
1.O
EO
low-frequency output deviation, for example at frequencies below 0.1 Hz, is due both t o temperature drift and t o internal transistor noise, so that it may be difficult t o distinguish between the two causes. It is rather important t o understand that noise and drift are two different effects, and as such they are listed separately by the manufacturer. Whatever the cause, any output-voltage deviation from the ideal can be “referred to the output”-as the expression goesand represented by the equivalent voltage-deviation generator, Do, in the equivalent circuit on the right in Figure 2. This is, however, still a n approximation t o the real amplifier not only because it has ideal input and output impedances, but also because we have omitted any representation for the “current drift” and “current noise” normally stated by the manufacturer. The term “referred to the output” means that all internal, random deviations from the ideal-either drift or noise-have been “lumped,” i.e., added, into a n equivalent deviation generator in series with the output terminals. By shorting the input terminals, this output voltage deviation may or may not be observable with a n oscilloscope, depending o n the operational-amplifier saturation levels. Effect of Output Deviation on General Feedback Amplifier. The effect of the output-voltage deviation, Do,o n the general feedback-amplifier may be analyzed with the equivalent circuit in Figure 3. In this circuit the point P is not available for physical measurement with a n oscilloscope probe, but Eo can be found indirectly as follows. When E1 equals zero-for instance by shorting the input physically-2, and 2 1 function as a voltage divider, assuming infinite input impedance, so that the voltage a t the summing point, S, is E -~
- 21
L1
+ ZfEo
O n the other hand, considering a loop through the operational amplifier,
Eo
= -AoEs
f Do
(2)
Eliminating the summing-point voltage, E,,
1.
1I
I
Near-real operational
.J
I
1
- A W - -i
Figure 3. General feedback-amplifier including open-loop deviation voltage, Do, “referred to the output.” Z , and Z , may be complex (reactive) impedances
if A,
f >> 1 + ZZI
which is then the output signal of the feedback amplifier when the input signal, E],is zero. From the superposition principle-applicable t o circuits with linear elements in general (5)-and the standard equation for the general feedback amplifier (3, the total output signal in the presence of a non-zero input signal is then n
/
7\17
11
+
-q Ao
For purposes of accuracy and flat frequency-response, operation is normally confined t o the range of frequencies where A,>> 1
+ Zf
(7)
2-1
so that the second bracketed term in Equation 6-the error factor-equals one. The resulting output signal with a nonzero input signal is then
1
(3) Zl
+ Zf
(9)
or better, in terms of the usual “error factor,” if A,
- 1 >> Lr - >> 1 21
Equation 8 shows-among other things-that the previously mentioned proportionality between Do and A, is very unANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
o
611
Figure 4. Equivalent circuit of voltage-follower with drift
1
I
I I
I
fortunate; otherwise the second term could be eliminated simply by making A , large enough. The Voltage Follower. As another example of a drift calculation, a similar analysis of the equivalent circuit for the voltage-follower in Figure 4 shows that
EI if A.
that the current drift commonly encountered in solid-state amplifiers could be completely ignored in the equivalent circuits. DISCUSSION
Equations 8 and 10 show that the parameter Do/A,-the open-loop output deviation divided by the open-loop gainoccurs quite naturally as a system parameter in the feedback amplifier. The physical meaning one may assign to this ratio is shown in Figure 6 by the two equivalent circuits for the operational amplifier. From a “black box” point of view, these two circuits are indistinguishable with respect to the output terminals. In the lower equivalent circuit any output deviation from ideality is “referred to the input”-as the manufacturer says-and the non-ideal input circuitry with the equivalent input-deviation generator, D,,is followed by a n ideal operational amplifier. The lower equivalent circuit has the advantage that the input voltage deviation, D,-be it due to drift or noise-can be compared quickly with the ideal summing-point signal. Furthermore, in quantization noise analysis of ac amplifiers, it is usual practice to refer noise sources to the input (4). The part of D,which is due to temperature or power-supply fluctuations is called “input offset voltage drift” and is given in microvolts/”C for temperature variations. For the supply voltage, it is given in microvolts per volts of power-supply variation. The term “input offset voltage” refers to the dc component of D, which must be balanced out somehow by applying a n external bias voltage in order to zero the dc output voltage. The part of D, which is due to the noise of the circuit components is called “input voltage noise” (BurrBrown Co.) or “voltage noise referred to input” (TeledynePhilbrick-Nexus Co.). In order to specify the signal deterioration in the resistivefeedback amplifier used here, we can formulate the closed-loop
+ Ds
>> 1 EXPERIMENTAL
A qualitative check on Equation 5 is summarized in Figure With the input grounded (E1 = 0) the closed-loop gain, 21/21,was varied and the closed-loop output voltage was recorded on a n XY-plotter. The closed-loop output voltage, E,, for the zero-signal condition is seen to follow Equation 5 in the sense that it increases when the closed-loop gain is increased, yet is not equal to zero even when the closed-loop gain is zero. The residual output voltage resulting when R,/RI = 0 can be used to obtain Do by simply multiplying it by the open-loop gain, A,, although it is easier t o use Equation 5 with R,/R, large enough to get a sizeable voltage swing. In this amplifier Do, “the drift referred to the output,” was about 10 volts peakto-peak. The data were taken on one of the two-tube operational amplifiers in the Heathkit analog computer. The drift in solid-state operational amplifiers is several hundred times smaller and more difficult to record, so that the top fluctuation in Figure 5 , which gives the factor D,/Ao directly, would be difficult to see on a solid-state amplifier. Furthermore, it is important to understand that this experiment was done on a vacuum-tube operational amplifier, so 5.
00
Figure 5. Closed-loop output drift and closed-loop gain (R,/R1) 0
I
0
I
I
40
I
80
I
I
120 t [secs]
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I
160
I
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200
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240
To obtain a n equation expressing the effect of drift o n a n integrator, Equation 8 is applicable. Taking the inverse Laplace transform after the substitutions 1.
I
I
,J
2f-t
1 -
sc
2 1+
R
we get
1
Eo=-AoE=-Ao E
-5I = - A o E , t D o AoJ
Figure 6. Equivalent circuits for the OP AMP with (a) voltage drift generator Do “referred to the output,” and (b) voltage drift generator D,? “referred to the input”
Figure 7. Magnitude of signal-to-drift voltage ratio os. closed-loop gain for the resistive-feedback amplifier output signal-to-drift voltage ratio (SDVR). Dividing the first term of Equation 8 by the second term and using the definition D, = D,/A,, we get
It is clear from the behavior of this equation (Figure 7)and perhaps even more from Equation 8 itself-that with a given input signal-to-drift ratio the best output ratio is obtained when R,/Rl >> 1. However, even then the “input drift” of the operational amplifier is amplified as much as the signal. When R,/R1>> 1, Equation 9, shows that the “input voltage drift generator” of the operational amplifier acts as though it were located in series with the input signal, El; however, this is not normally the location meant by the term “driftreferred to the input”, which is reserved for the location in Figure 6b immediately before the operational amplifier itself.
This agrees with another recent treatment (6) of the integrator which included not only voltage drift, but also current drift. The equation shows particularly how the “input offset voltage,” the dc component of ds,should be zeroed out by external bias adjustment; otherwise this dc component will be integrated, and the operational amplifier will eventually reach saturation. For those circuits requiring long integration periods or small input signals, the method of chopper stabilization is a valuable solution to the problem of voltage drift (5, 7). I n this paper we have only considered voltage drift and voltage noise in the analysis. Thus the lower model in Figure 6 is too simple for most solid-state amplifiers, since it does not include the output deviations due to the “input current noise” and “input bias current drift.” These input current deviations are represented by current generatosr-not voltage generators-in parallel with the input impedance in Figure 6. The resulting input circuitry, composed of both voltage and current generators, is a generalized method of representing amplifier non-idealities. As applied t o quantization noise in amplifiers this method of representation is in common use with audio and radio-frequency amplifiers ( 4 ) . Both input generators must be included in the analysis of solid-state operational amplifier circuits except, perhaps, the F E T operational amplifiers. Because of the theory of superposition, one would expect the effect of the input current generator to manifest itself by additive terms in the equations derived here. CONCLUSION
We have shown how, by the use of equivalent circuits for voltage drift in the operational amplifier, it is possible t o obtain over-all input-output transfer functions, like Equations 8 and 10. Such equations show how t o select amplifier signal levels and circuit parameters in order to combat the intrinsic deviations-drift or noise-in the real operational amplifier. Obviously, with large input signals, one need not be concerned about drift and noise, but our typical approach t o the simple circuits shows how to get a grasp of the random non-idealities in more complex circuits where the effect of both input voltage generators and input current generators must be assessed. RECEIVED for review May 20, 1971. Accepted September 22, 1971. (6) Larry L. Schick, ZEEE Spectrum, 8, 36 (1971 j. (7) E. A. Goldberg, RCA Reo., 11, 296 (1950).
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