H. V. Malmstadt and s. R. Crouch
University of Illinois Urbana, Nlinois
Systems for Automatic Direct Readout of Rate Data
The determination of the rate of change of one physical quantity with respect to another (often time) is an indispensable procedure in scientific and engineering investigations. For example, velocity, acceleration, temperature coefficient, and the rate of a chemical reaction are just a few of the important rate measurements. I n practice, the rates are usually graphically determined from a point-by-point or a recorded plot of the value of one quantity versus the other. However, this procedure is tedious, time consuming, and subject to bias, and it would be advantageous in many cases to obtain a direct readout of rate information. Unfortunately, the unwanted noise fluctuations associated with many typical experimental rate signals have made it quite difiicult to obtain the accurate direct readont of rate information without rather expensive circuitry. Consequently, the direct measurement of rates has not been widespread. It is the purpose here to present three systems for rate measurements that are based on comparison measurement principles. These systems are readily assembled from inexpensive commercial modules and a few parts, and, by virtue of the comparison technique, they provide high precision and accuracy. The first method is a manual balancing procedure that is presented to introduce the concept and principles of rate measurements by the comparison technique. It is capable of good results ss illustrated by experimental data, but it is also inefficient because of the slow response of the human servo in the detection and balancing steps. I n the second system, the human feedback loop is replaced with an inexpensive electmmechanical servo loop, and in the third system by an all-electronic feedback loop. These latter two systems are both reliable devices which should he generally applicable in research and instructional laboratories. The characteristics, limitations, and basic equations of these rate-measuring units are systematically developed from the most elementary circuit considerations. Experimental results are also presented for the application of the Direct Readout Ratemeter for automatic quantitative determinations based on initial reaction rates. In the final section, some of the classical electronic derivative systems are discussed to indicate limitations relative to the comparison procedures.
If the potential difference e,, is observed between points s and g, it is apparent that a t any time t the voltage e,, is equal to -ex plus the voltage drop (iRn) caused by the series current i through the resistor Rz, 1.e. a,, = iR2- e, (1)
Principles and Instrumentation
and since,
Assume that a certain physical quantity is changing value with respect to time, and that a transducer is available that produces an output voltage e, whose rate of change is directly proportional to the quantity to he measured. The main operation in the compari-
it follows by substituting for i i n eqn. (I),that
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son rate procedure is to use a standard slope generator whose rate of change of output voltage can be readily varied and adjusted to a known rate de,/dt that is equal to the unknown rate of change de./dt. A detector is necessary to indicate when the two rates are equal. If the detector is very sensitive and capable of discriminating against noise, the accuracy of the measurements will he primarily dependent on the accuracy of the standard slope generator. The specific con~parison techniques, slope generators, and detectors are presented in the subsequent sections. Rate Measurements by a Manual Comparison Technique
Principks. The first system to be described is based on the elementary circuit shown in Figure 1. The output voltage e. from the transducer is in series with the output voltage from the slope generator. In this case, the arrows through the symbols indicate that both output voltages can vary continuously a t specific rates, one determined by the variation of the physical qnantity to be measured, and the other by a specific switch setting on the standard slope generator. The resistors R, and Rz con~pletethe simple series circuit.
Figure 1:
Boric comparison circuit for rote meorvremenn
and if resistors RI and Rpare equal, it follows that
a t some time t. At some later time (t can be rewritten as follows
+ At) eqn. (4)
From eqn. (5) it is apparent that if the potential diierence e,, is to remain constant at every instant after time t, then it is necessary that changes Ae. in the unknown e, be balanced at each instant by equivalent changes Aes in the standard e,. This, of course, is the same as saying that the rate of change of the standard, de,/dt, must equal the rate of change of the quantity being measured, de,/dt. Note from Figure 1that when e, and ee,are measured against the common point (point g), their polarities are opposite, so that the balance condition is Note also from eqn. (5) that any potential difference between e, and e, a t time t will also remain a t every instant after time t if the rates are perfectly and continuously matched after time t. Detection of the Diffwmce Potential. It is assumed here that any desired rate can be obtained from the output of the standard slope generator by simply adjusting a dial or switches. There are limits to the rate adjustment for practical generators, as discussed in the next section. A good sensitive detector for this specific technique is the ordinary strip-chart potentiometric recorder, which is connected directly between points s and g, as illustrated in Figure 2. The schematic of the
Figure 2: Detection syskm for observing differences in rates b e t w e e n unknown and standard.
recorder is for the high-input impedance type such as the Heath EUW-20-A. As in all servo null-point potentiometers the reference voltage e, is automatically varied by the servo motor until i t is equal to the input voltage, which in this case is e,,, and the position of the pen on a chart is then proportional to e, and hence to the input voltage. As shown in Figure 2, a vibrating contact on an instrument chopper alternately samples the voltages e,, and e, at 60 cps or other suitable frequency.
If there is a difference between them, an ac voltage is produced which is amplified by the servo amplifier, and the amplified signal is used to operate the phase sensitive servo motor that is coupled mechanically to the variable reference source e,. The ac voltage on the control windings of the servo motor causes the motor to turn in the proper direction until e, equals e,,. Thus if connections are made as shown in Figure 2, the writing pen will continuously indicate the difference pot,ential e,, as a function of time.
Figure 3: Hypothetical balancing of standard and unknown rater by the manual comparison technique.
A typical rate measurement using a strip chart recorder to detect e,, is illustrated in Figure 3. Figure 3a shows the variation of the unknown potential e, with time. For clarity, e. is assumed to he changing a t a constant rate. At time to the output e, from the standard slope generator is zero so that the recorder pen indicates the variation of e, with time as shown in Figure 3c. At some later time t,, the standard slope generator, is switched to a specific output rate, and it is assumed in the example that de,/dt is initially less than that of the unknown signal as illustrated in Figure 3b. The recorder pen now indicates that e,, is still changing with time, although not so rapidly as at time to. At time tz the standard slope generator is switched to a higher rate of change, and its output slope is now greater than the unknown slope. The recorder pen reverses direction at t2 indicating that we have overshot the point at which the two rates are equal. Finally, a t time ts the standard slope generator is switched to a slightly lower rate, and it is noted from the pen indication (Fig. 3c) that the difference potential e,, remains constant with time. This indicates that the balance point has been reached where the rate of change of output voltage from the standard slope generator is the same as the unknown rate. Note from Figure 3c that a difference potential remains after time t3 even though the rates are perfectly matched. Hence, in contrast to many comparison Volume 43, Number 7, July 1966
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techniques, the detector does not have to indicate a true zero point when the slopes are balanced. Standard Slope Generator. The standard slope generator should be easily and rapidly adjustable to provide accurately known rates of change de,/dt in the range encountered in typical laboratory situ&ons. A useful range which a versatile slope generator should provide is from a few tenths of a mv per sec to a few v per see, a dynamic range of about 10" If the unknown rate lies outside this range, it is easier to amplify or divide the voltage rather than to design a generator with greater dynamic range. A very simple slope generator, which consists of a battery and a potentiometer with the slider driven by a motor, is illustrated in Figure 4. The output voltage is
de,=e, dl
R.C.
Eqn. (8) shows that the output rate of change of this slope generator can be conveniently varied by changing the input voltage e,. For example, if the integrator time constant (R,C,) is chosen to be 10 sec, and e, = 100 mv, the output rate of change will be 10 mv/sec. If e, is changed to 200 mv, the output rate will be 20 mv/sec. If this type of slope generator is used to provide the standard slope in the comparison rate procedure, the input voltage to theintegrator e,is directly proportional to the rate of change of the unknown voltage when the standard and unknown rates are equal, as can be seen by substituting eqn. (8) into the balance condition (eqn. (6)). de, -de, -e (at bslmce) - = - = 2 dl
dt
R.C.
(9)
The switch S, in Figure 5 is a reset switch, which is shorted to keep the output of the integrator zero until a measurement is made.
Figure 4;
A h p l e rtondard slope generator.
equal to iR, where i is a constant current and R, is a variable resistance. Therefore, the rate of change of output voltage de,/dt is determined by the rate at which the motor moves the slider. A wide range of output rates could be obtained by using a stepper motor, whose speed depcnds on the frequency of the voltage signal applied to the windings, to drive the slider. If this type of slope generator were used in the comparison rate procedure, the frequency necessary to drive the motor so that the output rate matched the unknown rate would be directly proportional to the rate of change of the unknown signal. Although such a slope genera, tor is not too practical at present, future developments in continuous potentiometers, stepper motors and inexpensive transistorized frequency dividers will probably make this standard slope generator desirable. An advantage of this type of slope generator is that a direct digital readout of frequency, and hence the rate of change of the unknown signal, could easily be made. A second type of standard slope generator, based on RC integration, is shown in Figure 5. In this system, which has been described in an earlier article ( I ) , the servo motor varies a standard potential ee,so that it is proportional to the integral of the input voltage e,. Eqn. (7) gives the relationship between e, and e,. (7)
For a given value of the input voltage e,, the output rate of change of the sewo integrator is given by 342
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Figure 5:
Servo integrator for generating the standard slope.
Although the servo integrator of Figure 5 can be used to provide an accurately known output rate of change, a more convenient slope generator is shown in Figure 6. This circuit is analogous to the servo system of Figure 5 except that an all electronic amplifier has replaced the mechanical linkages of the servo system. This amplifier is called an operational amplifier and, when connected as shown in Figure 6, provides an output voltage e, proportional to the integral of its input
Figure 6: dope.
Operotionol amplifier integrator for generating the standard
voltage e,. Eqn. (10) gives the output rate of change as afunct,ionof e,. de, - = - -e. dt R.C,
When this operational amplifier integrator is used to provide the standard slope in the comparison rate procedure, the voltage e, is directly proportional to t,he unlinown slope at balance. (at balance) e , = R E , de,/dt
(11)
Because the operatio~~al amplifier integrator is more convenient and compact t,han the servo integrator, it, was used as the standard slope generator in all of t,he systems described in this paper. Sperzfic Circuit f o ~Manual Rate Measurements. Figure 7 shows a practical circuit, composed of commercially available units and a few components, that was used for manual rate mcasurements. The standard slope generator was composed of two units; an operational amplifier (amplifier 1 of the Heath E W 19-A Operational Amplifier System) wired as an integrator, and a Voltage Reference Source (Heath E W 16) to supply the input voltage to the integrator.
rates of change were generated with a second operational amplifier integrator similar to the standard slope generator. With the standard slope generator properly biased, as discussed above, the input signal was connected as shown in Figure 7. The recorder was then turned on, and the standard slope was varied, by changing the voltage from the Voltage Reference Source, until the recorder indicated that the difference potential e,, was constant with time. The Voltage Reference Source setting, G, was then noted, and the experiment was repeated for different input rates. According to eqn. ( l l ) , the voltage e, should be directly proportional to the rate of change of the input signal, and a linear relationship between e, and de./dt was found over the range of input slopes studied (1 mv/sec to 100 mv/sec). A typical plot of the data
OPERATIONAL AMPLIFIER INPUT
REEORDER.
REFERENCE
e.
Figure 7:
Specific circuit for manual rate meosvremenlr.
The output from the Voltage Reference Source is continuously variable from 0.1 mv to 100 v. Since the time constant of the integrator (R,C,) was 10 see, the output from the standard slope generator was continuously variable from about 0.01 mv/sec to 10 v/ see. The numbered points on the operational aniplifier schematic in Figure 7 refer to specific pins on the front panel of the Heath system. Components were attached to these pins by means of clip-on connectors. To avoid errors in the standard slope, which might arise because of amplifier drifts and offset voltages. the integrator was frequently balanced by grounding the input and adjusting the amplifier bias until the output did not change with time. With the amplifier wired for integration, improper adjustment of the zero level and drift will cause an output slope even when the input voltage is zero. Since the accuracy of the rate measurement is primarily dependent on the accuracy of the standard slope, this adjustment was made prior to each measurement. The Heath EUW-20-A servo recorder was used to detect the difference potential e,,. The recorder sensitivity setting was 10 mv full scale, which was sufficient, to detect a few hundredths of a mv per sec difference in slopes between the unknown and standard. Results by the Manual Comparison Technique. To test the manual rate procedure, input signals of various
Figure 8: Plot of integrator input e, agoinst input slope for rnanuml rate rneosuremenh
for input signals of 10 to 70 mv/sec is shown in Figure 8. The rate results were reproducible to about 0,4y0 with this specific system. Automatic Rate Measurements by a Servo Comparison Technique
Manual adjustment of the standard slope generator, while capable of good accuracy and precision as showu in the preceding section, proved to be quite time consuming, and typically two to threeminutes of "hunting" were required before the exact voltage was found which would balance the two rates. Although this is not a serious disadvantage for unknown signals of constant slope, the manual system would be inconvenient and uureliahle for rate measurements with other types of signals. Even for a linearly varying signal, an automatic system that could instantaneously detect differences in the rates and adjust the standard slope generator mould greatly reduce t,he time required to make a measurement. Principles. The basic features of the automatic system that was developed can be understood by considering the functions of the human operator in the manual technique described above. The human operator must observe the difference potential e,, on a recorder, determine when there is a difference between Volume 43, Number 7, July 1966
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the unknown and standard rates, and adjust the standard slope generator until the difference or error disappears. I n this sense the human operator acts as a feedback loop, and he controls the standard slope generator to equal the slope of the unknown signal. The eye acts as a detector which senses the error (difference in rates) and sends this signal to the mind, which directs the body to vary the standard slope until the error signal disappears (slopes are equal). It is evident that the response time and accuracy are both dependent on human factors. The response time depends on how fast the body responds to the error signal and the amount of "hunting" to find the balance voltage. The accuracy depends on how small a difference in rates the eye can detect on the recorder and also on the quality of the standard slope generator.
Figure 9: Servo ly3tern for automatic rate rneorurementl by the cornpari.ion technique.
An electromechanical servo system for automatic rate measurements is illustrated in Figure 9. To continue the analogy to manual rate measurements, the automatic system replaces the eye and the mind (the difference detector and the controller) with a chopperamplifier. A chopper reed alternately malces contact with points s and g at 60 cps, and is connected to the input of a high gain amplifier. If there is a difference in potential between these two points, a square wave appears a t the amplifier input. This square wave is of the same frequency as the chopper (60 cps). The error signal is amplified by the servo amplifier and sent to the windings of the phase-sensitive servo motor. The servo motor and mechanically coupled potentiometer replace the body in the manual system. If there is a difference in potential between points s and g, the servo motor receives an ac voltage whose phase depends on whether point s is at a higher potential than point g or vice versa. The servo motor then turns a shaft coupled to a potentiometer and varies the input voltage e, to the standard slope generator in the proper direction so that the error signal will disappear. If a 344
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pen is linked to the servo-driven potentiometer, the position of the pen on a chart will be proportional to e,. Because the same basic circuit is used as in the manual system, the same elementary equations apply, and eqn. (11) is the balance condition for the automatic system as well. However, the servo system has the advantage in that the position of the pen on the chart is directly proportional to e, and hence to the rate of change of the unknown voltage. I t is important to note several differences between the manual rate system and the automatic system. Compare Figures 2 and 9 and not.? that in Figure 2 the recording potentiometer plays no role in changing e,. This must he done manually. The only function of the recorder in the manual system is to indicate the difference potential e,, as a function of time. In the automatic system shown in Figure 9, however, it is the servo motor which controls the rate of change of e, by controlling the input voltage e, to the integrator. Another important distinction between the two systems is that in manual rate measurements a true nullpoint between e, and e, is unnecessary. It is only necessary that these voltages be changing at the same rate. I n the automatic system, however, the magnitudes of the two signals, as well as their rates of change, must be equal at balance (the difference potential e,, must be zero). This requirement could lead to instability in the servo system and cause oscillation around the balance point until a true-null point was reached. Practical Servo Comparison Circuit. Specific details of the circuitry for the servo ratemeter are shown in Figures 10 and 11. Figure 10 is an over-all schematic of the Heath EUW-20-A servo recorder modified for rate measurements, and Figure 11 gives wiring details. The necessary modifications can he made on this recorder without altering any of the factory wired components. The standard "B" and "C" sockets in the rear of the recorder were removed, which allows access to the bridge and voltage divider circuits. Specific pins in the "B" and "C"sockets are labeled BI, B2, CI, C2,etc. Original testing of the circuit was done with clip-on components attached directly to these pins. A permanent circuit was later wired in a Heath EUW19-A-3 Adapter Chassis, which plugs on to the front of one of the Heath EUW-19-A operational amplifiers. No changes were necessary in the recorder "A" socket, but the normal connections are shown in Figure 10 to complete the diagram. By referring to Figure 9 and the general principles of servo rate measurements discussed in the preceding section, Figure 10 is easily understood. The standard slope generator is identical to that used in the manual system except that the integrator time constant (R,C,) was changed to 1 sec. The input voltage to the standard slope generator e, is obtained from the bridge circuit of the servo recorder. The actual bridge voltage is divided by a voltage divider composed of six precision resistors to get the proper input voltages for the integrator. Specific full scale spans of e, of 250 mv, 100 mv., 50 mv, 25 mv, and 10 mv may be selected with sensitivity switch 81, which switches in the proper divider resistors. Since the integrator time constant was chosen to be 1 sec, full scale values of e, may be read directly in terms of rates (250 mv/sec, 100 mv/
sec, 50 mv/sec, 25 mv/sec, and 10 mv/sec). In addition, the recorder's variable sensitivity knob may be used to obtain intermediate full-scaie spans. With these two sensitivity controls, a continuously variable full-scale span from about 3.3 mv/sec to 250 mv/sec may be obtained. A 2 mfd capacitor is placed across the sensitivity switch for damping purposes.
-
81
(%(Jig9
MOTOR A
I
I I
I
d
-
L Figure 10:
accurately known slopes from an operational amplifier integrator similar to the standard slope generator. Signals from 1 mv/sec to 100 mv/sec were applied. Figures 12 and 13 are recordings of data taken with the servo ratemeter. Recall that the pen deflection is directly proportional to e, and hence to the input rate. As expected, the servo balancing svstem tended to o s c i l l a ~ if not properly damped, and Figure 12 shows oscillations when the system is underdamped. 51 SENSITIVITY Direct proportionality beIN MV/SEC tween the pen deflection and the unknown rate may be seen in Figure 12 after the oscillationsceased. The exact time that the system came to balance was critically dependent upon damping in the recorder and the time constant of the standard slope generator. Figure 12 was obtained with the RECORDER BRIDGE recorder damping knob OFF. Even after oscillations stopped and the system seemed to be balanced, a damped oscillation would AT BALANCE begin again if spurious noises or mechanical movement of the pen occurred. An example of this may be seen in rate meosuremenh Figure 12 for the 100 mv/ sec simal. With urouer damping, the oscillations were rapiky damped asshoin by the recording in Figure 13. In this case the recorder damping was adjusted carefully until oscillations just ceased. Even with careful damping, small
%
I
O v e r - d l rchemdic of Heath EUW-2OA Recorder modified for
The modifications in the "B" socket of the recorder disconnect the internal voltage divider and allow use of the external divider discussed above. The recorder sensitivity switch must be turned to EXTERNAL when the servo ratemeter is attached. The changes in the "C" socket reverse the polarity of the output voltage from the recorder's bridge circuit. It is necessary to reverse the polarity of e, so that an unknown signal whose rate of change is positive will result in a positive e, and thus a negative standard slope, i.e., de,/dt = -de,/dt. In Figure 11 a wiring diagram of the ratemeter is shown. The part of the circuit shown in dotted lines was wired in the adapter chassis. Connections in the socket marked "0.A." refer to specific pins in the 5-pin female socket in the chassis which plugs on to the 5-pin male socket of the operational amplifier. Connections are also shown for the 5-pin female "B" and "C" sockets which ~ l u on e to male sockets a t the rear of the recorder. The - - .....- - ....- .- - - - .- - - - - - - - - - - ."B" socket was attached to the adapter Figlure 11: SpeciRc wiring details of the servo ratemeter. chassis by means of 3-stranded cable. A seoarate "C" socket. wired as shown in fluctuations occur around the balance point, and ini~i'gure11,was used tb replace the standard socket in the tial pen overshoot is noticeable in Figure 13 when the recorder. rate is suddenly changed. Results by the Servo Comparison Technique. The The direct proportionality between e, and the nnservo ratemeter was tested by applying input signals of A
-
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known rate for two different full-scale settings, 50 mv/ sec and 25 mv/sec, is also shown in Figure 13. Note the very fast response time of about 1 sec for the servo ratemeter when compared to the 2 or 3 min of hunting required for the manual system.
these changes in the control circuit, the same basic comparison circuit is used. The basic principles of the all-electronic device are illustrated in Figure 14. As before: the standard slope generator is illustrated as a battery with an arrow through it which indicates
Figure 13: Recordings of rote measurements by properly damped servo ratemeter at two different full scale renritivitier lo1 50 mv/sec f d l rcole. ( b l 2 5 mv/sec full scale.
that the output can vary continuously at selected rates. However, as shown by the lower figure in dotted lines, it is composed of an integrator and an inverter. The function of the inverting amplifier xi11 Figure 12: Recordings of rote mearuremenh by an underdmmped servo system. Input rates .re from 0-1 00 mv/sec.
Automatic Rote Measurements by on All-Electronic Comparison Technique
The aut,otnatic servo ratemeter gives satisfactory results for input signals of constant slope and for nonlinear signals which change rates rather slowly, but the finite response time of the electromechanical system (about 1 sec for full-scale response) is a limitation for faster varying rates. Also, the careful damping required to keep the system from oscillating might prove inconvenient for routine use with noisy signals. The all-electronic system described in this section can overcome some of these limitations by replacing the mechanical linkages of the servo unit with an electronic control amplifier capable of extremely fast response. I n addition, the all-elect,ronic unit has provision for controlliilg the response time so that response is no faster than needed, which is valuable for work with noisy input signals. Principles. The basic features of the all-electronic ratemeter are similar to those of the servo unit. I n place of the chopper-amplifier, servo motor, and mechanical couplings, an electronic amplifier designed for feedback control is used. The operational amplifier is ideally suited for this purpose. Except for 346
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-
,._._...___.__.__
READOUT
Figure 14:
Boric circuit of the all-electronic comparimn ratemeter.
be discussed later in this section. As with the manual system (Figures 1 and 2), the unknown signal e, and the output e, from the standard
slope generator are arranged in a simple series circuit. The difference potential e,, is the input to the control operational amplifier (OA-3). The operat,ional amplifier feedback serves to force the two aniplifier inputs to very nearly the same potential. The feedback loop in this case is from the amplifier output through the slope generator, inverter, and resistor R2 to point s (one of the inputs). If the second input (point g) is ground, the other input (point 8 ) is forced by the feedbacl~to be very close to ground potential (called avirtnal ground j. The amplifier output voltage t,hat is necessary to maintain e,, near zero volts is called e, as before. From Figure 14 and previous considerations of the comparison circuit it can be seen that if the amplifier maintains e,, very near zero v at all times, the rate of change of e, and e, must be equal at all times. Because the basic circuit is identical to that used in the manual and servo devices, the equations for balance are the same. However, as was the case with the servo ratemeter, it is necessary to invert t,he polarity of the feedback voltage to ensure that the unknown and standard change potential in opposite directions. Recall that in the servo system this was done by reversing the connections in the bridge circuit of the recorder. In the all-electronic unit it is necessary to add an additional operational amplifier to iuvert the signal. The inverting amplifier may be placed either before the standard slope generator or after it as shown in Figure 14. If the inverter is placed after the standard slope generator, the balance voltage e, is negative when the inuut sloue is positive so the balance condition is
of the currents flowing towards a junction is zero. In this case point s represents the junction, and the input current i is shown to divide into currents il and i2. It is arbitrarily assumed that de,/dt is positive, and the resulting polarities of e, and e, are indicated in Figure 15. No assumption will be made as to the direction of current flow, as this depends on the relative magnitudes of the three potential sources. By Iiirchoff's first law we may write i il z; = 0 . . t = - < I - z2 (13)
+ +
The potential difference e,, is seen to be equal to any of the battery voltages at a given time t minus the voltage drop across the resistance in series. If consideration is given to the signs of these potentials, the following expression may be written: ( e , ) , - i R1
-(e,), -(e,L
=
e,,
- 71R3 = e,, - ilRj = esn
These expressions may be solved for the currents by remembering that the operational amplifier maintains e,, very near zero volts, so that i
=
e,/Rl
i,
=
e,/Ra
zl = e,/R,
Substitution of these expressions into eqn. (13) gives the following expression fore,:
Since e, changes with time we may differentiate eqn. I t is important to note that in the all-electronic ratemeter an external recorder or voltmeter must be used for the readout of the balance voltage e,. In the servo unit the readout is an integral part of the device. This operational amplifier circuit is similar in principle to the differentiators used in analog computers (8. Noise Suppression in the All-Electronic Unit. Although the circuit diagramed in Fignre 14 will give an arcurate measure of the rate of a synthetic signal, the response is so rapid that the rate of change of any superimposed noise will also be measured. A simple method for controlling the response speed of the system is shown in Figure 15. AresistorRr is placed around the control operational amplifier. The addition of Rr Figure 15: Boric drcuit showing noire ruppresion rerirtor Kj. lo1 over-oll circuit, lbl simplified e q u i r creates a second feedback network .lent.irc"it. for the amplifier, and the diagram at the bottom of Figure 15 shows a sirn(14) to obtain the rate of change of e., i.e., plified equivalent circuit composed of batteries and &stors.- This circuit may bevery simply solved, and the resulting equation will indicate the effect of R, on the resnonse meed. To solve the circuit. Kirchoff's Recall that eqn. (10) gives the expression for de*/dt first law ;s used,'which states that the aliebraic sum Volume 43, Number
7,July 1966 / 347
when the operational amplifier integrator is used as the standard slope generator. Substituting eqn. (10) into eqn. (15), and remembering that the polarity of the de,/dt was inverted by the inverting amplifier, yields
the integrating capacitor so that the output of the standard slope generator can be returned to zero between measurements.
Eqn. (16) is now easily solved to give the balance voltage e, in terms of the rate of change of the unknown signal de,/dt. Rearrangement of eqn. (16) gives
R&C, de,
=
-Rl dt
(17)
Eqn. (17) may be readily integrated, taking the limits of integration to bee, = 0 a t t = 0 and e, = e, a t t = t . The result of this integration is
If, as before, we choose resistors R1 and R2 to be equal, then eqn. (18) reduces to Figure 16:
Eqn. (19) predicts that the addition of R, causes e, to approach balance exponentially, and a t balance e, is proportional to the rate of change of e,. Note that the time it takes for e, to reach its final value may he calculated from eqn. (19) for any value of R, (the time necessary for the exponential term to become much smaller than 1). Also note that when the exponential term becomes insignificant equ. (19) reduces to eqn. (12), which indicates that the addition of R, merely reduces the response speed of the circuit and does not change the basic relationship. All Electronic Ratemeter
Specific Circuit. Figure 16 shows the detailed circuit of the all-electronic system. The Heath Operational Amplifier System (EUW-19-A) was used to construct the circuit. The finished circuit was wired in a blank adapter chassis (Heath EW-19-A-1) which plugs on to the front panel of the amplifier system. Three ampliiers are used in the ratemeter. Amplifier 1 is wired as an integrator and generates the standard slope. Although the exact component values for the standard slope generator are not highly critical, their choice is governed by considerations of offset drifts a t the amplifier input. These drifts, usua.lly a few mv with unstabilized amplifiers, can result in large errors in the standard slope as previously mentioned. To minimize the effect of drifts a t the input of amplifier 1, components were chosen, so that the input voltage to amplifier 1 is much larger than the expected drifts. A time constant (R,Cs) of 50 sec was chosen for the standard slope generator. With this time constant the input to amplifier 1 will be 50 mv if the unknown rate is 1mv/sec. A shorting switch is connected across 348
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Journal o f Chemical Education
SpeciRc circuit of all-electronic robmeter.
Amplifier 2 is the inverting amplifier whose necmsity has been previously discussed. Amplier 3 is the control amplifier. Switch S2 in the feedback loop of this amplifier allows the selection of a variety of R, values. With the specific resistors shown in Figure 16 the response speed of the system may be varied from a few milliseconds to several seconds. Switch position 1 contains no resistance, and in this position the system responds with its normal unmodified speed. In addition, two front panel jacks allow external resistors of diierent values from the fixed resistors to be placed in the feedback loop when position 1 is selected. A 500 K potentiometer at the output of Amplifier 3 is used to display a fraction of the balance voltage e, on a convenient recorder or meter. Function switch S1 is a 5-position switch which consists of two wafers, S l a and S l b . Positions 3-5 are used to balance the three operational amplifiers. For example, in position 3 the input to the ratemeter is grounded and the output of Amplifier 2 appears at the output jack. The bias for Amplifier 2 may then be adjusted until its output is zero. Positions 4 and5 allow similar balancing of the other two amplifiers. The switch Sw across the integrating capacitor should be shorted when the amplifiers are being balanced. In position 1 of switch S1 the normal input jack is used for connecting the unknown signal. Position 2 has the same function except that the auxiliary input jack is connected to point s. These two inputs allow signals from two different sources to be measured without disconnecting leads. In actual applications of this system i t has been advantageous to connect a standard sweep generator to the auxiliary input jack to calibrate the entire system. Applying a standard
linear sweep enables the proportionality constant between the output voltage and the input slope to be determined experimentally. This constant includes the fraction of e, which is displayed on the meter or recorder.
adjustment, rebalancing of the amplifiers was not necessary for periods up to three weeks as long as the amplifier system was not turned off. The effect of the feedback resistor R, on the response time of the ratemeter is illustrated by the recorded curves in Figure 18. When a feedback resistor is present, the approach to balance is exponential as predicted by eqn. (19), and the balance voltage is independent of RI. Accurate measurements of the response time as a function of R , were made oscilloscopically and are compared in Table 1 with the theoretical response time calculated from eqn. (19). To obtain Table 1.
Effect of Feedback Resistance on Differentiator Response Time
10 &leg 4 . 7 Mee 1.0 ~ e g 470 K 300 K 100 K
Figure 17: Recordings of rote measurements with all-oiectronic ratemekr. Input rotet are from 1-1 0 mv/rer
Results. The rate measuring unit of Figure 16 was tested by applying synthetic signals of known slope from an operational amplifier sweep generator. The balance voltage e, was recorded by connecting a strip chart recorder to the output jack. A typical recording of e, values for input slopes of 1 to 10 mv/sec is shown in Figure 17. The reproducibility and stability of the measuring system may be seen from the duplicate sets of determinations. Because of the closed nature of the feedback network and the careful choice of components
Figure 18: Effect of noise suppression resistor Rl on balancing time of ollelectronic rotemeter. Valuer of R1 ore given above each recording.
to minimize the effects of amplifier drifts, balancing of the operational amplifiers was not critical. I n most cases the amplifiers were balanced once after the initial 30-min warm-up period. Except for this initial
60 msec 130 msec 540 msec 1.2sec 1 . 9 see 5 . 6 sec
5 6 . 5 msec 120 msec 565 msec 1.20 sec 1.88 see 5.65 sec
the experimental data, a 100 mv/sec input signal was applied, and the time necessary for the output voltage e, to reach 63% of the final balance voltage measured. The theoretical time constant from eqn. (19) is RzR,C. tas. = -RI where R1 = 10 K, and R, = 4.7 Meg. The value of C, was found experimentally to he 12 mfd by measuring the rate of discharge of C, through a precision resistor. This was necessary for accurate comparisons because the nominal value of C, was 10 mfd 20'3. The agreement between the theoretical and exper~mental response times indicates that eqn. (19) describes the behavior of the system. Note that eqn. (19) can be used to calculate the value of Rlwhich will give a preselected response speed. Figure 18 also illustrates the effect of Rl on the amount of noise appearing in the output of the ratemeter. Longer response times lead to significant noise suppression. This effect is better illustrated in Figure 19 in which a noisy input signal was applied to the ratemeter. As can be seen from the recorded trace of the input signal e,, the noise superimposed on the linear voltage sweep was cyclic and of rather low frequency. Note that the instantaneous rate of change of e, differed rather drastically from the average Changing RI to 100 I< decreases quite significantly the ability of the system to follow the noise. Note, however, that when R, is 100 K it takes about 30 sec before the balance voltage can be read on the recorder. Thus for a given input signal a compromise must be reached between the amount of noise which can be tolerated and the desired response time of the system. In most cases it is desirable to decrease the response by the minimum amount necessary to read the balance voltage accurately. Another alternative which Figure 19 suggests, is to use a noise averaging readout system for e,. For example, all three curves may be easily averaged by eye. For extremely noisy signals, however, eye integration is more difficult and electronic averaging Volume 43, Number 7, July 7 966
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349
techniques are capable of more precise results. By electronically integrating e, for a short time interval, a fast response of the ratemeter can be used with little sacrifice in accuracy and precision. A simple unit for averaging such signals as those shown in Figure 19 will be described in a later article.
Glucose
+ 02 Oluoose
oaidase
Gluconic acid
+ HzO?
This reaction may be easily followed by observing concentration changes of the products from a secondary reaction which proceeds at a much faster rate than the above primary reaction. For example, if the H202 oxidizes iodide to triiodide in the presence of a catalyst, the triiodide concentration may be followed by a spectrophotometric, potentiometric, or amperometric technique.
I n all of the methods of following the reaction an electrical signal is obtained whose initial rate of change is directly proportional to the glucose concentration (4-6). I n this study spectrophotometric measurements were used to follow the reaction because of the many problems associated with accurate rate measurements from spectrophotometric curves. I n many cases output signals from an expanded-scale spectrophotometer are noisy, and it is difficultto determine the slopes graphically. If the Direct Readout Ratemeter with noise suppression could be applied to spectrophotometric signals, much operator time would be saved in routine analyses. The Experiment
I
TIME
I
The oxidation of glucose mas followed spectrophotometrically by measuring the absorbance of triiodide at 360 nip. A block diagram of the complete reaction rate measuring system is shown in Figure 20. The Spectro
Figure 19: Effect of noisy input signal on rotemeter output for three values of noile suppression resistor Rf.
Application t o M e a s u r e m e n t of t h e Initial Rates of Chemical Reactions
Analytical methods which make use of the measurement of the rates of chemical reactions have become important in recent years for the quantitative determination of many substances. For the highest reproducibilty in using kinetic methods, it is important to measure initial reaction rates. I n most cases a transducer is available that produces an output voltage e , whose rate of change is directly proportional to the concentration of some species in solution. I n this section the application of the Direct Readout Ratemeter to these important rate measurements is described. Experimental results are presented for the quantitative determination of glucose by a spectrophotometric rate procedure. These results indicate that the Ratemeter presented herein may be directly applied to this method to obtain accurate and precise initial rate data that are directly related to concentrat,ion. Pardue and Dahl (3) have used an all-electronic system for potentiometric rate measurements. However, the specific circuit described by these authors is not appliable to grounded signal sources such as the output from some spectrophotometers. Enzymatic Determination of Glucose
The specific enzymatic determination of glucose is an important analytical method that utilizes rate measurements. The method is based on the use of the enzyme glucose oxidase to catalyze the following reaction: 350 / Journal of Chemical Education
Figure 20: Block diagram of automatic rpectrophotometric reaction rote meowing system.
unit was the Spectro section of the Spect,ro-Electro titrator (E. H. Sargent and Co., Chicago, Ill.). A phototube transducer mas used (GE 929) in place of the photoconductive detector supplied with the instrument. The thermostated reaction cell has been previously described (4). For this study, water at 30 + 0.l0C was pumped through the cell jacket. The 525 mp interference filter on the titrator was used with a visible cutoff filter (Corning No. 5860) t,o isolate a narrow band of radiation near 360 mp. The current from the phototube was the input signal to the recording photometer, which was recently described (I). The voltage output from the photometer e, was the input to the ratemeter of Figure 16. The output of the ratemeter e, was recorded on a standard servo recorder. All reagents and standards were prepared as previously described (4). Glucose standards were in the range of 20 to 80 ppm. P~oeedure. The operational amplifiers were balanced after allowing about 30 min for warm-up. For this work a feedback resistor of 100 K was selected to minimize
noise fluctuations. For calibration purposes the glucose standard of highest concentration (usually 80 ppm) mas first used, and the 500 K calibrating potentiometer on the ratemeter was adjusted to give nearly fullscale deflection on the recorder. For direct readout purposes a glucose standard of 50 ppm was used, and the calibrate knob adjusted so that the pen deflected exactly 50 divisions on t,he 250 n ~ vrecorder scale. After a suitable warm-up period for the Spectro unit (about 30 min) 3 ml of a composite solution containing enzyme, buffer, catalyst, and iodide was pipetted into the thermostated reaction vessel, and the stirring was begun. The recording photometer was switched to a full-scale sensitivity of 200 X 10-'o amps, which corresponded to about 15% Transmittance full scale. The pen on the photometer was adjusted to about 30 divisions with the Standardization knob to allow for the small decrease in absorbance due to dilution when the glucose was added. To initiate the reaction, 1.00 rnl of the glucose standard was injected into the cell with a syringe. The reaction was allowed to proceed until a good average value of the rate mas ohtained on the recorder. Because of a short induction period in the glucose reaction (l(t30 see) the actual measurement time was 'hout 4&50 sec. However, readings of the rate were obtained within 1Ck 15 sec after the reaction assumed first-order kinetics.
p 00 PPM
r 50 PPM
Figure 21: Recorded rate curves and direct rsodoul of slopes for glucose concentrations of 50 and 8 0 pprn.
shown in Figure 21. Noise fluctuations in the input signal to the ratemeter were often quite severe and necessitated the use of a slow response time. These fluctuations arose from several sources. Because the sensitivity of the recording photometer was very high, small fluctuations in the lamp intensity in the single beam Spectro unit appeared as noise in the photometer output. It was also difficult at this sensitivity to eliminate fluctuations due to particles and air bubbles which eutered the light path while the solution was being stirred. To minimize these effects, the stirring unit was usually turned off after an initial mixing period of 10-20 see. Even in the presence of such noise sources, recorded slopes such as those shown in Figure 21 were obtained and are easily averaged by eye. As previously mentioned, electronic integration of the ratemeter output for a short time interval would he advantageous for eliminating the eye averaging process. Table 2 shows a typical set of results obtained for aqueous glucose samples. Recorded values of the rates were reproducible within about 2Yw A plot of Table 2.
Reproducibility of Results for Aqueous Glucose Solutions
Glucose concentration
Recorded slope chart divisions
Itel. rtd. dev. $
the recorded rates versus glucose concentration is linear and passes through the origin, which permits a direct readout of glucose concentration after calibmtion of the ratemeter with a single standard. These results indicate that the Direct Readout Ratemeter can he applied directly to spectrophotometric reaction rate procedures. One of the advantages of this method over some rate measuring devices is that the slope of the rate curve may be recorded as a function of time. Thus, if a reaction begins to deviate from its initial rate as a reactant is consumed, the recorded rate will change. Direct recording of the rates of chemical reactions might also prove to be of importance in basic kinetics investigations for determining the order of a reaction. The Direct Readout Ratemcter is now being used in student experiments basrd on rpaction rate procedures similar to that for glucose. Details of the sperifir experiments will be reported iu future articles. Classical Rate Measuring Devices
I n this section, some of t h classical ~ electronic rat? measuring circuits are presented and discussed in relation to the comparison techniques introduced in this paper. These devices are operational amplifier units which provide output voltages proportional to the derivative of the input voltage. Differentiator8 with intentionally reduced frequency response are also discussed. Operotionol Amplifier Differentiotors
Results. Typical recordingsof the rate curves and the measured slopes for two glucose concentrations are
The classical textbook operational amplifier (7) for measuring the rate of change of an input signal el is Volume 43, Number 7, July 1966
/ 351
shown in Figure 22. The amplifier inputs are labeled s and g, and, as mentioned before, the amplifier feedback serves to maintain these two inputs at very nearly the same potential. Since point g is ground, the amplifier maintains point s a t virtual ground. If it is assumed that negligible current flows into the amplifier itself, which is usually the case, the current supplied by the input signal must flow past point s and through Rl as indicated in the diagram. This input
Figure 22:
to decrease the frequency response of the unit. Figure 23 shows a diierentiator with reduced frequency response. If components are chosen so that R,C, = R&', this differentiator is said to provide maximum accuracy up to a cut-off frequency which is related to R'C, by,
where f,,is the cut-offfrequency (2).
Clasrical operational amplifier differentlotor.
current is simply the rate of change of charge with respect to time, i.e., i = dQldt
(20)
where Q is the charge in coulombs on the capacitor. The capacitance Ct is defined as the ratio of the charge Q on the capacitor to the potential across the capacitor. This potential must be the full input voltage el since point s is nearly ground potential. Hence we may write, and,
The amplifier output voltage e, must be such that it will maintain this current. Since the entire voltage e, appears across the feedback resistor Rf if point s is nearly ground, we may write, e.
=
-iRr
and, substituting i from eqn. (21) gives, e, = -RJC,(deildt)
(2.1)
(23)
Thus the output voltage is proportional to the derivative of the input voltage. Within the limitations imposed by the frequency response and gain of the operational amplifier and the maximum output voltage which the amplifier can supply, the circuit in Figure 22 gives a true derivative of the input signal. As such it is useful in applications in which input signals are available with little or no superimposed noise fluctuations. The gain of a true differentiator must increase in direct proportion to the frequency of the input signal, since the rate of change of the input signal is proportional to the frequency. Hence the classical true differentiator is often avoided in experimental measurements where input signals are often associated with noise. Noise Suppression in Differentiators. In order to use an operational amplifier differentiator to measure the rate of change of a noisy input signal, it is necessary 352 / journal of Chemical Education
Figure 23: Operational amplifier differentlotor with decreased frequency response.
A qualitative understanding of the operation of this modiied differentiator can be obtained by considering the parallel combination of RfCfin the amplifier feedback loop. This parallel combination divides the current i into two components i , and iz as shown in the diagram. At low frequencies the reactance of the capacitor, X , = 1/27rfC1, is very high, and most of the current goes through the resistive path. Thus for low frequencies the circuit acts like the unmodified differeutiator. At high frequencies, however, the capacitive reactance decreases so that much of the current is shunted around the resistor. This makes the circuit of Figure 23 an integrator for high frequencies, and the gain falls off. Discussion. From the principles of classical rate measuring devices in the preceding section, it is apparent that there are many problems associated with the direct measurement of the rates of slowly changing signals in the presence of noise. Often this noise is of rather low frequency (a few cps), and even frequency degraded differentiators may give noisy output signals that are often useless. Attempts to reduce the frequency response of the classical diierentiator to almost zero lead to basic difficulties, and extensive filtering networks are often used to rid the signal of noise components. The basic problem is one of selectively amplifying signals near zero frequency in the presence of higher frequency noise. Conclusions
The data and circuits in the preceding sections indicate that the direct readout of rate information can be made with high accuracy and precision by the comparison technique. These procedures have significant advantages in measuring the rates of typical experimental signals. By virtue of the comparison principle, the accuracy is limited chiefly by the accuracy of a standard slope generator. Because the slope generator produces a signal to match the unknown signal, it is
possible to make the slope generator respond only to slow changes in the unknown. If a detector is available with noise the comparison technique may offer further noise suppression. Literature Cited
A. Philhrick Researches, Ine., Boston, Mass., 1960, p. 12. (3) P.
