The simplest case is illustrated in the following example. The equivalent circuit in Figure 1 represents an electrochemical cell connected to a potentiostat capable of controlling the potential between the reference and working electrodes Ectl. In all electrochemical cells. there is some resistance between the tip of the reference electrode and the outside of the double layer. This resistance is referred to as the uncompensated resistance. As current flows through the electrochemical cell. a potential is developed across the uncompensated resistance. This potential is calculated by:
When the uncompensated resistance gets high (as it does in pure water or organic solvents) or the current becomes large, a significant potential error may occur. Using this simple model, the control potential is related to the potential across the double layer (the desired control potential) by this equation:
The Positive Feedback and Current Interrupt methods of iR Compensation are implemented in the Model 273 and are discussed in the following sections.
.......Eapp = the desired control potential.
Substituting (Edl + Eerr) for Eecl (from equation 2) yields this new equation:
1) The cell electrodes are adjusted and their positions fixed.
2) The cell is filled with the solution of interest and the potential is adjusted so that there is no ... electrochemical reaction.
3) A 50 to 200 Hz square wave of small amplitude (approximately 50mV) is applied and the ... current waveform is monitored. See the Model 273 manual for details
. The design of the feedback circuit and the speed of the potentiostat determine the maximum possible ( value. In many cases 100% feedback ("a" = 1) is impossible; severe system ringing or oscillation can prevent the potentiostat from maintaining control of the applied potential.
Setting the amount of positive feedback can be a confusing task, largely because the various types of experiments demnd different degrees of compensation. In this light, two broad classes of experiments can be identified: "DC" experiments and "Fast Response" experiments.
DC experiments are those in which the potential at the electrode is changed slowly. One example of this involves the following steps: first you step the potential by a small amount, next you allow the transient effects of the step to subside; and finally you measure the steady state (or DC) current. This type of experiment is common in corrosion laboratories. In a DC experiment, it is practical to compensate for nearly 100% of the potential error, since the current measurement is made after the transient effects have subsided.
In Fast Response experiments. the potential is changed rapidly either in a ramp or in rapid steps. These experiments require some subjective judgement. An example of this class of experiment is Potential Step Chronoamperometry. In this method, the chemist is interested in the shape of the cur rent vs. time curve after applying a large potential step. If you compensate for more than about 85% of the potential error, you may get ringing in the current vs. time curve. If ringing occurs, you cannot make a valid current measurement until it subsides.
(Note: The adjustment of the current range and the feedback network takes several milliseconds. Thus, in very fast experiments, the automatic current range feature should not be used)
Figure 3 shows a block diagram of the positive feedback scheme in the Model 273 In this system, the adjustment of the feedback is controlled by a multiplying digital-to-analog converter (or MDAC). The output of the MDAC is a percentage (from 0.025% to 100% in 0.025% steps) of the input voltage (in this case the output of the current-to-voltage converter) The ratio of the positive feedback and the reference electrode resistors is 2 to 1. This makes the effective correction signal range from 0.05% to 200% of the feedback signal (in 0.05% steps).
For example. when you select the 100 mA range, the maximum uncompensated resistance which can be offset is 20 Ohm (see Table 1). You can't compensate for an Ru value larger than 20 Ohm. However, this maximum Ru value will Generate a potential error of 4 V (20 Ohm x 200 mA).
Table 1 indicates the range and resolution of the correction for each current range. "Correction range" is defined as the maximum resistance which may be offset. "Resolution" is defined as the smallest increment in which the resistance can be programmed.
Let us continue with this example to see how errors due to exceeding the resolution limit can occur when you change the current range. If the current range changes to 1 mA during the experiment. (assuming a resistance resolution of 1Ohm), the resistance actually used for correction will change to 1234 Ohm. The improved resolution allows the Ru value that is actually used for correction to be identical to the entered Ru value.
A problem may occur when you shift to a more sensitive range. For instance. if you were to shift to a current range of 10 uA (and thus a resolution limit of 100 Ohm, the resistance actually used for correction will be 1200 Ohm. This is true because the value can only be represented to the nearest 100 Ohm. In other words, the error will now increase to 34 Ohm . Similarly, a further shift to the 1 uA current range would create an actual resistance of 1000 Ohm (a 234 Ohm error!) and a shift to 100 nA would create an actual resistance of 0 Ohm (a 1234 Ohm or 100% error!). Generally speaking, as the current range shifts to less sensitive ranges, the error will be accordingly reduced.
While it is important to understand the resolution of the feedback path, the maximum potential error that this represents is only 2 mV. In the above example. using the 10 uA range, the resistance error is 34 Ohm. However, the error at this resistance is not catastrophic, even at the maximum current of 20 uA, the error is only 0.62 mV.
1) Correction is continuous.
2) Correction is effective even using the fastest scan rates possible with the Model 273.
1) The adjustment of the feedback is tedious and subjective.
2) The feedback reduces the stability of the potentiostat and may lead to severe ringing or oscillation.
3) Because the feedback signal affects system stability, it is normally not possible to adjust the potentiostat so that "a" = 1. The user normally has to be satisfied with a potential error correction of "a"= 0.65 to 0.9.
4) When you make the initial feedback adjustment, you are assuming that the uncompensated resistance is constant. This can be an incorrect assumption. If Ru does vary. you will not be completely compensating for it.
1) The desired potential is applied under control of the microprocessor (using DAC 1).
2) At a carefully selected time. the switch at the output of the power amplifier (PA) is opened, causing current flow in the cell to go to zero.
3) Figure 5 is an ideal view of the reference electrode potential plotted vs. time. When the switch is opened, the error (Eerr) in Figure 1 will become zero, since no current is flowing. Thus. Eref is controlled by the potential across the double layer (Edl).
4) After the switch is opened. the potential across the double layer is determined by the rate of discharge of the cell. The rate of discharge is the product of the double layer capacitance (Cdl ) and the general faradaic impedance (Zf). The potential will vary exponentially.
5) The Model 273 measures Eref, every 5usec and stores the resulting data in a high speed memory. After 32 measurements the switch is closed and the potentiostat regains control of the cell. The total off time for each interrupt is less than 200 usec.
6) Once the connection to the cell has been re-established, the microprocessor reads the data out of the high speed memory and selects two data points (dependent on the current range of the I/E converter). It then calculates the intercept of the straight line through these two points at time zero (see below for details). The intercept point gives the value of the potential error (Eerr) in the cell.
7) The Eerr signal (up to +/- 4096 mV at 2 mV resolution) is then applied to the cell using DAC 2. This correction cycle takes approximately 100 usec after the cell is reconnected.
8) Finally, if a potential scan is underway, DAC 1 updates the control potential.
Note that this technique measures the potential error (the quantity of interest), and not the uncompensated resistance.
There is also an initial deviation to consider. When the correction potential is first applied, the current may increase, thus increasing the potential error. However, the next current interrupt cycle will detect this additional error and correct it. These interations will continue for several cycles, until the error caused by the correction itself is smaller than the resolution of the correction.
In all cases, you can calculate the uncompensated cell resistance using Ohm's Law (E = iR) if you know the potential error and the current flow at the instant of interest. However, when you make this calculation, you must determine your values carefully. Ohm's Law dictates that at small current values, small changes in potential error represent large changes in uncompensated resistance. At one microamp, a change of one millivolt represents a one kilohm change in uncompensated resistance!
For these reasons, you should choose your two data points carefully, keeping in mind these two important considerations:
1) The first data point must not be distorted by the cell turn-off.
Both points must be chosen early enough in the decay to ensure that the straight line approximation is valid.
From experience, we have determined a set of default data point times that are used with the Model 273. Table 2 shows the values we have chosen for each current range. These values may be modified from the front panel of the Model 273. See the Model 273 manual for further details.
Current.........................First Point.....................Second Point
Range............................Time.............................Time
1 A, 100 mA....................10us................................20us
10 mA to 100 nA..............75us..............................150us
TABLE 2. Default values for data point times.
During operation, the timing of the current interrupt is determined by a value programmed from the front panel . This value can be set to any value between 4 msec and 30 sec in multiples of 4 msec (the time base of front panel operation). If the entered value is not a multiple of 4 msec, the computer will round to the nearest 4 msec multiple. When no scan is applied, the correction cycles will occur under control of the front panel setting. (The default setting is one cycle per second.)
The selected scan rate determines the rate at which the potential is updated. If the scan rate is less than 62.5 mV sec, the potential is updated in 0.25 mV steps. The processor determines how many 4 msec intervals will occur between each step.
For example. using a 10 mV/sec scan rate, the potential should be updated every 25 msec. Since the time base is 4 msec, the update will occur three consecutive times at 24 msec and a fourth time at 28 msec. The 24 msec cycle will be divided into six 4 msec cycles. Five cycles will occur in which the current and potential are measured, but the potential is not changed. On the sixth cycle, the current and potential will be measured, and the potential will be changed by 0.25 mV.
While the scan is in progress. the interrupt cycles are coupled with the potential update cycles. When the time between interrupts has elapsed, the interrupt is executed during the next data acquisition and potential update cycle.
Thus, while the potential is being scanned, each data acquisition and potential update cycle will occur in the following sequence:
1) The current will be measured.
2) The current interrupt cycle be executed.
3) The potential correction signal will be applied using DAC 2 (see Figure 4) 4) The applied potential will be updated using DAC 1.
Some care must be exercised in the timing of the interrupts. As the number of interrupt cycles increases, the percentage of time that the system is not controlled increases. Also, because of the nature of the measurement, the system noise will increase. However, the larger the time between interrupt cycles, the larger the uncorrected error may become. This is especially true if the current is changing very rapidly. For example, in a corrosion experiment, the current can change by an entire order of magnitude during one cycle (when the scan is within a few millivolts of the corrosion potential). Thus, if the interrupt is occuring every 10 mV, the current interrupt may cause erroneous potentials to be applied to the cell.
Moreover, this system is really a positive feedback system with a long time constant (minimum of 4 msec) and a complex transfer function. Thus, special care must be exercised when choosing values which cause current interrupts to be widely spaced. For slow scans. interrupts occurring at one second intervals should yield good results.
The upper trace shows the electrometer voltage waveform and indicates a potential error of approximately 1485 mV. The lower trace shows the I MONITOR output at a current of 85 mA. When full scale current output occurs (on any scale), the I MONITOR potential is always 1 V. Thus, using the 100 mA current range, an output potential of 850 mVindicates a current of 85 mA. Using the potential error and current values. you can calculate the uncompensated resistance using Ohm's Law. In this case, Ru is approximately 175 Ohm.
In this example, the potential correction is 1485 mV. In order to maintain - 935 mV across the double layer, it is necessary that the control potential be set at -2420 mV.
(Note: The value displayed on the front panel of the Model 273 can be either the actual control potential or the value of the voltage across the double layer.)
In many cases, especially when interrupts are performed frequently, you may have to use a modified reference electrode which has a better high frequency response. In this case a platinum wire is placed at the tip of the reference electrode and coupled to the reference electrode using a small capacitor. 1, 2 This allows the reference electrode to operate at high frequency and to respond to rapid changes in potentials.
2) Current Interrupt will correct for any changes in the uncompensated resistance as the scan progresses.
3) You have no adjusting to do. AII you have to do is press a button to enable the correction.
2) If the correction is not updated frequently enough, the applied correction can be in error, although the error will not be as great as when no correction is made.
3) In some cases, this technique can cause the entire system to oscillate.
2. Hermann. C.. et al. Anal. Chem.. 40. (1968). p. 1173.